TW 


•• 


PHYSICS 


FOR 


UNIVERSITY   STUDENTS 


BY 

HENRY    S.    CARHART,    LL.D. 

PROFESSOR    OF    PHYSICS    IN   THE    UNIVERSITY   OF   MICHIGAN 


PART    I. 

MECHANICS,    SOUND,    AND     LIGHT 


Boston 

ALLYN     AND     BACON 
i  896 


Copyright,  1894 
BY    HENRY    S.    C  ARM  ART 


PRESS  OF 

-Korfeiotli  antr 

BOSTON,  U.S.A. 


PREFACE. 


THE  prevailing  practice  in  giving  instruction  in  Physics 
to  undergraduate  students  in  American  colleges  and 
universities  is  a  judicious  combination  of  the  text-book  and 
the  lecture  systems.  This  is  particularly  true  for  a  first 
course  covering  the  entire  subject,  and  laying  a  foundation 
of  general  principles  for  more  advanced  study  by  special 
courses  and  laboratory  work.  This  practice  the  author 
has  followed  in  teaching  large  classes  for  many  years  ;  but, 
finding  no  book  which  meets  his  needs,  he  has  felt  impelled 
to  prepare  his  own  text.  The  present  volume  is  an  exten- 
sion of  one  written  several  years  since  for  the  sole  use  of 
his  classes.  Fellow  teachers  into  whose  hands  the  earlier 
book  has  fallen  have  encouragingly  advised  the  expansion 
of  it  into  a  volume  of  somewhat  less  modest  pretensions, 
for  the  benefit  of  others  who  employ  similar  methods. 
The  result  is  a  text-book  and  not  a  treatise  on  Physics. 
No  attempt  has  been  made  to  secure  completeness.  The 
book  is  not  a  cyclopedia  of  Physics.  Only  such  topics  have 
been  selected  as  appear  most  important  from  the  point 
of  view  of  a  general  survey  of  the  science  ;  and  an  effort 
has  been  made  to  place  them  in  a  logical  relation  to  one 
another.  Somewhat  more  attention  has  been  given  to 
Simple  Harmonic  Motion  than  is  customary  in  an  ele- 
mentary course.  Its  extensive  application  in  the  study 
of  alternating  currents  of  electricity,  added  to  its  earlier 


IV  PREFACE. 

important  service  in  Mechanics,  Sound,  and  Light,  renders 
a  more  thorough  study  of  it  imperative. 

The  book  is  not  intended  to  take  the  place  of  the  living 
teacher.  It  leaves  room  for  the  personal  equation  in  in- 
struction ;  but  it  will  relieve  the  student  of  a  large  part  of 
the  labor  of  taking  notes ;  and,  it  is  hoped,  will  secure  for 
him  more  accurate  statements  than  he  would  be  likely  to 
obtain  from  listening  to  lectures  without  the  aid  afforded 
by  a  text-book  of  principles. 

In  many  cases  the  method  of  the  calculus  has  been  em- 
ployed without  its  formal  symbols.  The  course  in  Physics 
represented  by  this  book  is  supposed  to  precede  the  study 
of  the  calculus,  and  the  methods  used  will  prepare  the 
student  for  the  employment  of  that  branch  of  mathematics 
in  more  advanced  courses 

Free  use  has  been  made  of  the  books  referred  to  in  the 
headings  of  articles,  and  especially  of  Violle's  admirable 
Oours  de  Physique. 

The  present  volume  covers  the  work  done  in  the  first 
course  extending  over  one-half  of  the  academic  year.  The 
second  part  will  be  devoted  to  Heat,  Electricity,  and  Mag- 
netism. 

The  author's  thanks  are  due  to  Assistant  Professor  J. 
O.  Reed  for  many  valuable  suggestions  and  for  careful 
reading  of  the  proof  sheets. 

A  few  of  the  cuts  in  Sound  and  Light  were  kindly 
furnished  by  the  publishers  of  Anthony  and  Brackett's 
Physics,  with  the  permission  of  the  authors. 

UNIVERSITY  OF  MICHIGAN,  September,  1894. 


\ 


CONTENTS 


MECHANICS. 

CHAPTER  PAGE 

I.  Introduction  ..........         1 

II.  Kinematics     ..........         8 

III.  Kinetics .42 

IV.  Kinetics  (Continued} 66 

V.  Mechanics  of  Fluids                                                                                99 


SOUND. 

VI.     Nature  and  Motion  of  Sound        ......     137 

VII.     Physical  Theory  of  Music    .         .  176 


LIGHT. 

VIII.     Nature  and  Propagation  of  Light 226 

IX.     Reflection  and  Refraction 241 

X.     Dispersion      ..........  285 

XI.     Interference  and  Diffraction         .         .         .         .         .         .  297 

XII.     Color 308 

XIII.     Polarized  Light 317 

INDEX  337 


REFERENCES. 


The  letters,  enclosed  in  brackets  accompanying  the  headings 
of  articles,  refer  to  the  following  books,  numerals  denoting  pages : 

A.  and  B.,  Anthony  and  Brackets  Text-Book  of  Physics. 

B.,  Barker's  Physics. 

Bl.,  Blaserna's  Theory  of  Sound. 

D.,  Danieirs  Text-Book  of  the  Principles  of  Physics  (Second 
Edition) . 

H.,  Helmholtz's  The  Sensations  of  Tone,  translated  by  Ellis;  the 
second  numerals  in  small  brackets,  the  German  text  of  Die  Lehre 
von  den  Tonempjindungen. 

K.,  Koenig's  Quelques  Experiences  d'Acoustique. 

L.,  (in  Mechanics)  Lodge's  Elementary  Mechanics. 

L.,  (in  Light)  Lommel's  The  Nature  of  Light. 

M.  and  M.,  Maxwell's  Matter  and  Motion. 

P.,  Preston's  Theory  of  Light. 

S.,  Spottiswoode's  Polarization  of  Light. 

T.  and  T.,  Thomson  and  Tait's  Elements  of  Natural  Philosophy* 

T.,  Tait's  Light. 

Tyn.,  Tyndairs  Sound. 

V.,  Violle's  Cours  de  Physique. 

Z.,  Zahm's  Sound  and  Music. 


MECHANICS 


CHAPTER   I. 


INTRODUCTION. 

1.  Physics  Defined Physics  formerly  comprised  the 

study  of   all  the  phenomena  of  nature.     In  more  recent 
times  it  has  abandoned  the  history   of  organized  life  on 
the  one  hand,  and  the  study  of  celestial  phenomena  on  the 
other ;  and  it  is  now  restricted  to  the  study  of  the  general 
properties    of    matter,    particularly   to   those    phenomena 
which    involve    changes   in   the    energy    associated   with 
matter,    without    altering    the    essential    constitution   of 
bodies. 

The  real  nature  of  matter  is  unknown  to  us.  We  know 
something  about  its  properties,  that  it  is  conserved  in 
quantity,  that  it  apparently  occupies  space  and  affects 
our  senses.  It  is  preeminently  the  vehicle  of  Energy,  the 
capability  which  one  body  possesses  of  producing  motion 
or  effecting  changes  in  another  body. 

Modern  Physics  is  the  exposition  of  those  phenomena  of 
nature  which  involve  corresponding  changes  in  the  asso- 
ciated energy.  Formerly  it  was  largely  a  study  of  matter ; 
now  it  is  largely  a  study  of  energy. 

2.  Boundaries  of  Physics.  —  It  must  not  be  assumed 
that  Physics  is  separated  from  related  sciences  by  sharply 


2  MECHANICS. 

defined  boundaries.  It  overleaps  the  "  metes  and  bounds  " 
set  for  it  in  scientific  classification  and  enters  the  domain 
of  its  nearest  neighbors.  Since  all  natural  phenomena  in- 
volve energy,  and  Physics  is  largely  a  study  of  energy,  it 
is  evident  that  every  natural  science  furnishes  abundant 
material  for  physical  investigation.  A  science  is  classified 
in  accordance  with  the  chief  aims  to  which  it  is  devoted. 
Thus  Chemistry  studies  the  atomic  organization  and  struct- 
ure of  the  molecule.  It  endeavors  to  ascertain  how  the 
so-called  elements  enter  into  the  composition  of  both  or- 
ganic and  inorganic  bodies,  and  the  laws  governing  all 
those  inner  changes  in  matter  which  affect  its  properties 
and  identity.  The  combustion  of  carbon,  the  rusting 
of  iron,  the  burning  of  limestone,  the  fermentation  of 
wine,  are  changes  involving  the  constitution  of  the  mole- 
cules of  the  several  bodies.  They  are  therefore  chemical 
changes.  At  the  same  time  they  involve  energy  changes, 
and  are  therefore  appropriate  studies  for  the  physicist.  In 
fact,  there  are  so  many  problems  in  which  Physics  and 
Chemistry  are  equally  interested  that  a  new  subdivision  of 
science,  called  Physical  Chemistry,  is  even  now  in  process 
of  differentiation.  So  also  a  large  part  of  modern,  astron- 
omy is  so  much  concerned  with  the  physical  constitution 
of  celestial  objects,  with  their  intrinsic  and  variable  bright- 
ness, with  the  self-luminosity  of  suns  and  the  panorama  of 
nebulous  condensation,  that  it  has  won  for  itself  the  title 
of  Physical  Astronomy,  or  Astro-Physics. 

Mental  Philosophy  and  Natural  Philosophy,  or  Physics, 
were  once  both  characterized  by  the  employment  of  the 
metaphysical  method.  The  adoption  of  the  inductive 
method  and  the  experimental  plan  of  attack  in  Physics  led 
this  branch  of  study  far  away  from  its  ancient  ally ;  but 
now  Psychology  has  appropriated  the  methods  of  Physics, 


8 

and  the  two  are    moving   along  the  converging  lines  of 
experimental  research. 

But  energy  may  be  transferred  from  one  body  to  an- 
other, and  may  be  transformed  into  the  various  forms  which 
it  is  capable  of  assuming,  without  any  changes  in  the  con- 
stitution of  matter.  Physics  has,  therefore,  a  province 
independent  of  chemistry,  physiology,  and  other  allied 
branches. 

3.  The  Method  of  Physics.  —  A  science  is  character- 
ized no  less  by  the  method  essential  to  it  than  by  its  aim. 
The  aim  of  Physics  is  to  ascertain  the  causal  connection 
between  related  phenomena.  Such  study  is  justified  by 
the  accumulated  experience  of  the  human  race,  that  under 
the  same  conditions  the  same  results  flow  from  the  same 
causes.  In  other  words,  the  events  of  nature  are  not 
fortuitous.  This  principle  is  embodied  in  the  expression, 
the  constancy  of  the  order  of  nature. 

But  to  ascertain  the  causes  of  physical  phenomena  and 
the  laws  of  action  of  physical  forces,  it  is  absolutely  essen- 
tial that  the  consequences  of  any  provisional  theory  of 
them  should  be  subjected  one  by  one  to  the  control  of  ex- 
perimentation. In  experiment  the  phenomena  are  produced 
under  conditions  controlled  by  the  operator.  In  this  way 
the  necessary  relationships  are  established,  and  the  results 
flowing  from  a  single  cause  are  separated  from  all  others. 

Causal  relations  suggested  by  an  attentive  observation 
of  phenomena,  and  supported  by  reasoning  and  experi- 
ment, must  in  the  end  be  subjected  to  the  ultimate  test  of 
measurement.  Modern  Physics  is  essentially  quantitative 
in  character.  It  is  not  enough  to  know  that  a  relation 
exists,  but  that  relation  must  be  expressed  numerically. 


4.  Fundamental  Units  of  Measurement.  —  Since  all 
the  phenomena  of  nature  occur  in  matter,  and  are  pre- 
sented to  us  in  time  and  space  relations,  physical  measure- 
ments involve  the  choosing  of  three  fundamental  units, 
viz.,  the  unit  of  length,  the  unit  of  mass,  and  the  unit  of 
time.  This  particular  selection  is,  however,  a  matter  of 
convenience  rather  than  necessity,  and  rests  upon  several 
considerations  which  properly  determine  the  choice  of 
these  fundamental  quantities. 

All  other  units  employed  in  Physics  are  denned  in 
terms  of  those  of  length,  mass,  and  time.  They  are, 
therefore,  called  derived  units,  in  distinction  from  the 
other  three,  which  are  called  fundamental  units. 

The  three  fundamental  units  now  universally  employed 
in  Physics  are  the  centimetre,  the  gramme,  and  the  second  ; 
and  the  system  of  measurement  founded  upon  them  is 
called  the  C.G.S.  or  absolute  system. 

5.  The  Unit  of  Length.  —  The  centimetre  is  the  hun- 
dredth part  of  the  length  of  a  bar  of  platinum  at  0°  C., 
preserved  in  the  national  archives  at  Paris,  and  known  as 
the  metre  des  archives.  It  was  constructed  in  accordance 
with  a  decree  of  the  French  Republic  passed  in  1795 
on  recommendation  of  a  committee  of  the  Academy  of 
Sciences. 

The  value  of  the  metre  in  the  earlier  feet  of  different 
countries  is  as  follows : 

Foot  in  Metres.  Metre  in  Feet. 

France 0.3248394  ....  3.078444 

Austria 0.3161109  .     .     .     .*  3.163446 

Prussia  and  Denmark 0.3138535  ....  3.186199 

England  and  Russia 0.3047945  ....  3.280899 

Baden  and  Switzerland  0.3000000  3.333333 


INTRODUCTION.  5 

Foot  in  Metres.  Metre  in  Feet. 

Sweden 0.2960010  ....  3.368126 

Hanover 0.2920947  ....  3.423547 

Bavaria 0.2918592  ....  3.426310 

Hesse 0.2876991  ....  3.475854 

Wiirtemberg 0.2864903  ....  3.490519 

Saxony 0.2831901  ....  3.531197 

By  Act  of  Congress  of  the  United  States  in  1866  the 
metre  was  defined  to  be  39.37  inches. 

The  metre  was  intended  to  be  the  ten-millionth  part  of 
an  earth-quadrant  from  the  equator  to  the  pole.  It  is  now 
known  that  such  a  quadrant  is  about  10,002,015  metres. 
This  difference  between  the  ideal  and  the  legal  metre  illus- 
trates the  difference  between  a  theoretical  and  a  practical 
unit. 

6.  The  Unit  of  Mass.  —  Mass  is  the  quantity  of  matter 
in  a  body.  The  unit  of  mass  in  the  C.G.S.  system  is  the 
gramme.  Theoretically  it  is  the  mass  of  a  cubic  centimetre 
of  distilled  water  at  the  temperature  of  maximum  density, 
or  4°  C.  Practically  it  is  the  l-1000th  part  of  a  standard 
mass  of  platinum  preserved  in  the  archives  at  Paris  and 
called  the  kilogramme  des  archives.  The  theoretical  and 
practical  definitions  again  turn  out  to  be  not  absolutely 
identical.  National  prototypes  of  the  metre  and  the  kilo- 
gramme, made  by  an  International  Commission,  are  preserved 
in  the  Bureau  of  Weights  and  Measures  in  Washington. 

The  gramme  was  selected  as  the  unit  of  mass  because  of 
its  convenience,  since  it  is  nearly  the  mass  of  a  unit  vol- 
ume of  water  at  maximum  density ;  and  as  water  is  taken 
as  the  standard  in  determining  specific  gravity,  it  folloAvs 
that  density,  or  the  mass  of  matter  in  a  unit  volume,  and 
specific  gravity  are  numerically  equal. 


6  MECHANICS. 

7.  The  Unit  of  Time.  —  The    unit  of   time   universally 
employed  is  the  second  of  mean  solar  time.     An  apparent 
solar  day  is  the  interval  between  two  successive  transits  of 
the  sun's  centre  across  the   meridian  of  any  place.     But 
the  apparent  solar  day  varies  in  length  from  day  to  day 
throughout  the  year  by  reason  of  the  varying  speed  of  the 
earth  in  its  orbit.     Hence  the  mean  or  average  length  of 
all  the  apparent  solar  days  throughout  the  j^ear  is  taken. 
This  is  divided  into  86,400  equal  parts,  each  of  which  is  a 
second  of  mean  solar  time. 

8.  General  Subdivisions  of  the  Subject.  —  It  is   con- 
venient to  make  two  general  divisions  of  the  subject-matter 
of  Physics : 

1.  Physics   of    Matter.      This    includes    the    laws    of 
Motion,  the  mechanics  of  Solids  and  Fluids,  and  Sound. 

2.  Physics  of  the  Ether.     The  ether  may  be  a  refined 
kind  of  matter.     At  all  events,  it  is  desirable  to 'consider 
by  themselves  those  branches  which  deal  particularly  with 
the  ether,  and  which  are  not  completely  explicable  with- 
out taking  this   medium   into    account.     They  are   Heat, 
Light,  Electricity,  and  Magnetism. 

That  branch  of  the  first  general  subdivision  which  inves- 
tigates the  action  of  force  is  called  by  most  logical  writers 
Dynamics.  It  is  commonly  called  Mechanics,  a  term 
employed  by  Newton  to  designate  the  science  of  machines 
and  the  art  of  making  them. 

Since  force  is  known  only  by  the  motion  it  produces,  a 
discussion  of  the  laws  of  pure  motion  should  precede  that 
of  the  laws  of  force.  This  constitutes  the  subject-matter 
of  Kinematics.  To  the  idea  of  space  involved  in  geometry 
is  added  in  this  branch  that  of  time. 


IN  TR  OD  UCTION.  1 

Dynamics  is  divided  in  accordance  with  the  two  methods 
in  which  force  is  recognized  as  acting,  viz. : 

a.  As  preventing  motion  or  change  of  motion. 

b.  As  producing  motion  or  change  of  motion. 

Dynamics,  therefore,  includes  Statics,  in  which  equilib- 
rium under  the  conjoint  action  of  two  or  more  forces  is 
considered ;  and 

Kinetics,  in  which  the  relation  of  forces  to  motion  is 
studied. 

Our  chief  attention  will  be  given  to  Kinematics  and 
Kinetics. 


MECHANICS. 


CHAPTER   II. 


KINEMATICS. 

9.  Motion  (M.  and  M.,  38).  —  The  configuration  of  a 
system  of  points  or  minute  material  particles  is  their  relative 
positions.  When  a  change  of  configuration  is  considered 
only  with  respect  to  its  state  at  the  beginning  and  end  of 
the  process  of  change,  without  respect  to  time,  it  is  called 
displacement. 

But  when  the  attention  is  directed  to  the  process  itself, 
as  taking  place  within  a  certain  period  and  in  a  continuous 
manner,  then  the  change  of  configuration  is  called  motion. 

Motion  is  the  change  in  the  relative  position  of  a  mate- 
rial particle. 

A  material  particle  is  supposed  to  be  without  dimensions, 
but  to  possess  all  the  properties  of  matter. 

Kinematics  is  the  science  of  motion  considered  in  itself 
and  apart  from  the  causes  producing  it.  Ampere  separated 
it  from  rational  mechanics  as  a  branch  by  itself,  and  gave 
to  it  the  name  which  it  now  bears. 

It  is  of  consequence  to  observe  that  all  motion  is  merely 
relative,  since  there  are  no  fixed  points  in  space  to  which 
absolute  motion  can  be  referred.  Maxwell  says :  "  Any 
one  who  will  try  to  imagine  the  state  of  a  mind  conscious 
of  knowing  the  absolute  position  of  a  point  will  ever  after 
be  content  with  our  relative  knowledge." 


KINEMATICS. 

10.  The  Path.  —  The  succession  of  positions  of  a  mate- 
rial particle  is  called  its  path  or  trajectory.  The  path  of  a 
material  particle  must  always  be  continuous.  There  can 
be  no  abrupt  change  of  velocity  or  of  the  direction  of  mo- 
tion ;  such  abrupt  change  would  imply  the  action  of  an  in- 
finite force.  By  abrupt  change  is  meant  one  occurring  in 
zero  time.  Two  consecutive  portions  of  the  path  of  a  point 
would  then  come  together  at  an  angle.  Such  a  change 
would  require  an  infinite  force.  But  if  the  angle  is 
rounded  off  with  a  small  arc,  then  time  is  involved  in  the 
change  and  a  finite  force  may  effect  it. 

If  any  point  of  a  path  be  selected,  the  particle  will  pass 
through  that  point  at  least  once  during  the  motion.  It 
may  pass  through  it  more  than  once  if  the  path  is 
curvilinear. 

Mathematical  curves  may  be  discontinuous,  but  a  ma- 
terial particle  cannot  traverse  a  discontinuous  path  while 
retaining  continuous  existence  in  time  and  space. 

11.  Direction  of  Motion  and  Curvature  (T.  and  T.,  2). 
-  The  direction  of  motion  at  any  point  of  a  curved  path  is 
the  geometrical  tangent  to  the  curve  at  the  point.  The 
direction  of  the  motion  then  changes  from  point  to  point 
along  the  curve.  The  rate  of  change  of  direction  per  unit 
length  of  the  curve  is  called  the  curvature.  This  may  be 
either  constant  or  variable.  It  is  constant  in  a  circle  or  a 
helix ;  it  is  variable  in  an  ellipse  or  a  parabola.  The  cur- 
vature of  a  straight  line  is  zero. 

The  curvature  at  any  point  of  a  curve  is  the  reciprocal 
of  the  radius  of  the  circle  which  most  nearly  coincides  with 
the  curve  at  the  point.  This  may  be  demonstrated  as 
follows : 


10  MECHANICS. 

Consider  the  curvature  between  two  contiguous  points 
P  and  Q  (Fig.  1).  Let  6  be  the  angle  be- 
tween the  two  radii  PO  and  QO,  and  let  s 
be  the  arc  PQ.  Then  6  is  also  the  angle 
between  the  tangents  at  P  and  $,  and  this 
angle  is  the  entire  change  in  direction  in 
passing  from  one  point  to  the  other.  The- 
curvature  is  therefore  6/s.  Let  r  be  the 
radius.  Then 

rO  =  s, 

and  ?-  =  !.  (1) 

s       r 

If  the  two  points  are  indefinitely  near  together,  r  is  the 
radius  of  the  osculating  circle.  The  curvature  is  therefore 
equal  to  the  reciprocal  of  the  radius  of  this  osculating 
circle. 

In  the  case  of  a  circle  the  angular  change  in  direction  in 
going  once  round  is  2?r.  The  curvature  is  this  angle  di- 
vided by  the  arc  traversed,  or  by  Znrr.  Therefore  the 
curvature  for  a  circle  is 

27T 


12.  Speed  and  Velocity  .  —  Time-rate  of  motion,  with- 
out reference  to  the  direction  of  motion,  is  called  speed. 
Thus  we  speak  of  the  speed  of  a  horse?  the  speed  of  a  can- 
non ball,  the  speed  of  a  railway  train. 

But  when  the  motion  is  a  directed  quantity,  that  is, 
along  a  definite  line  whose  direction  is  given,  the  rate  of 
motion  is  then  called  velocity. 

When  the  material  point  traverses  equal  spaces  in  equal 
times  along  a  right  line,  it  describes  uniform  rectilinear 


KINEMATICS.  11 

motion.     The  velocity  along  the  line  is  the  constant  space 

traversed  in  the  unit  of  time. 

0       4  _  ¥-  _  *     Let    the    'distance     OM  -  =  s 
•  (Fig-  2)  be  described  in  the 

time  t,  this  distance  being  measured  along  the  path  OX 
from  the  point  0,  where  the  point  is  at  the  time  t  =  0. 


From  which 


or  .  the  velocity  is'  the.  constant  ratio  of  the  space  to  the 
time  employed  in  describing  it.  It€may  be  represented 
geometrically  by  the  path  OA  traversed  in  the  unit  of  time. 
Both  the  space  s  and  the  time  t  may  be  reduced  to 
infinitesimal  values  without  affecting  their  ratio.  Call 
the  infinitesimal  space  ds  and  the  corresponding  time  re- 
quired to  describe  it  dt  ;  then 

ds 

v  =  _  —  . 

dt 

If  the  motion  is  rectilinear  but  not  uniform,  let  J^Tand  Mf 
(Fig.   3)  be  the  positions    of 

the  material  particle  at  the  °  MM  _  A  X 
two  neighboring  epochs  t  and 

t'.     Then  vl  —   ;  —  -  is  the  mean  velocity  during  the  time 

t1  —  t.  It  is  the  constant  velocity  with  which  a  particle, 
travelling  along  the  right  line  OX,  would  describe  the 
space  MM'  in  the  time-interval  t'  —  t.  But  it  is  not  the  in- 
stantaneous velocity  at  the  time  t  .  Imagine  now  MM'  to  be 
reduced  to  infinitesimal  dimensions,  that  is,  to  tend  toward 
a  value  of  zero;  and  let  its  value  then  be  represented  by  ds 
as  before.  Also  let  the  time  of  traversing  this  minute 
distance  be  dt.  Then  precisely  as  before 

'  .  * 

dt 


12  MECHANICS. 

Geometrically  this  velocity  may  be  represented  by  the 
line  MA  laid  off  from  M  toward  X.  The  velocity  at  the 
time  t  is  the  distance  which  the  body  would  describe  if  it 
should  continue  to  move  uniformly  from  that  instant  for  a 
unit  of  time. 

The  practical  unit  of  velocity  is  the  velocity  of  one  cen- 
timetre per  second. 

13.  Acceleration.  —  When  a  particle  traverses  a  right 
line  with  variable  velocity,  a  case  of  special  interest  pre- 
sents itself  where  equal  changes  of  velocity  take  place  in 
equal  intervals  of  time.  The  motion  is  then  uniformly 
accelerated  (or  retarded),  and  the  constant  change  of 
velocity  in  the  unit  of  time  is  called  the  acceleration.  Ac- 
celeration is  the  time- rate  of  change  of  velocity.  Let  v0 
be  the  initial  velocity  at  time  0,  and  let  v  be  the  final 
velocity  at  time  t ;  also  let  a  be  the  acceleration.  Then 
from  the  definition 


(3) 

whence  v  —  v0  +  at (4) 

If  the  time  t  is  reduced  to  the  infinitesimal  value  dt 
and  the  corresponding  velocity-change  is  dv,  then 

dv 

*=*• 

This  expression  is  called  the  derivative  of  the  velocity 
with  respect  to  the  time.  It  applies  equally  well  to  either 
uniform  or  variable  acceleration. 

The  practical  unit  of  acceleration  is  an  acceleration  of 
one  centimetre  per  second  per  second ;  that  is,  it  is  a 
change  in  velocity  of  one  centimetre  per  second  taking 
place  in  a  second  of  time. 


KIN  EM  A  TICS. 


13 


Acceleration  may  be  either  positive  or  negative.  A 
negative  acceleration  is  a  retardation. 

PROBLEMS. 

1.  If  a  body  start  from  rest  with  a  uniform   acceleration   of 
2  metres  per  second,  find  its  velocity  at  the  end  of  3  minutes. 

2.  A  body  starts  with  a  velocity  of  300  metres  per  second.     If  it 
comes  to  rest  jn  1^  minute  2.5  seconds,  find  the  uniform  negative 
acceleration,     j  *o         v  (fe^.,,£< 

3.  A  body  has  an  initial  velocity  of  6  metres  per  second ;  find  its 
velocity  at  the  end  of  1,  2,  3,  and  6  seconds  respectively,  if  a  equals 
9.8m. 

14.  Space  described  in  Uniformly  Accelerated  Mo- 
tion.—  In  uniformly  accelerated  motion  the  equivalent 
velocity,  or  the  velocity  with  which  a  uniformly  moving 
particle  would  describe  the  same  space  in  the  same  time, 
is  equal  to  the  half  sum  of  the  initial  and  final  velocities. 
Hence 


8  = 


(5) 


Geometrically  this  may  be  represented  by  means  of  a 
right  triangle  (Fig.  4),  in  which  the  vertical  lines  drawn 
at  small  equal  distances  rep- 
resent the  velocities  at  suc- 
cessive instants  of  time. 
These  velocities  form  an  ar- 
ithmetical progression,  since 
the  variation  is  constant.  Let 
the  equal  divisions  along  the 
base  of  the  triangle  be  the 
small  time-interval  dt.  Then  if  the  particle  had  started 
from  rest  at  0,  the  mean  velocity  would  be  represented 
by  the  line  ab  midway  between  0  and  M,  and  this  equals 
half  the  final  velocity  MN.  But  if  the  initial  velocity  is 


14  MECHANICS. 

AB,  then  the  mean  equivalent  velocity  is  cd,  the  line 
drawn  midway  between  A  and  M.  But  cd  is  equal  to 
the  half  sum  of  AB  and  MN,  or  the  mean  velocity  is 


ds 

Moreover,  each  instantaneous  velocitv  -  -  multiplied  by 

at 

the  small  time-interval  dt,  during  which  the  velocity  may 
be  assumed  to  be  constant,  is  ds,  the  small  space  described 
during  time  dt.  This  space  is  one  of  the  narrow  strips 
making  up  the  triangle.  Hence  the  entire  space  described 
during  the  time  t  is  numerically  equal  to  the  area  of  the 
figure  AMNB,  which  is  the  expression  in  equation  (5). 

If  we  substitute  in  equation  (5)  the  value  of  v  obtained 
from  equation  (4),  we  have 

t=*'  +  *.t  =  vj  +  laf-  ...    (6) 

Multiply  together  (3)  and  (5)  and 

v2  -  v* 
as=    _^L, 

or  v*  =  v02  +  2as  ......      (7) 

If  the  initial  velocity  is  zero,  then  (4),   (6),  and  (7) 

become 

v  =  at       ......     (8) 

s  =  lat*    ......     (9) 

v*=2as  .......  (10) 

When  t  is  unity  (9)  becomes 

*  =  J«, 

or  the  space  traversed  in  unit  time  is  half  the  acceleration. 

If  t  is  unity  in  (8)   the  velocity  equals  the  acceleration. 

Therefore    the    space    described   during  the  first   second, 

when  the  body  starts  from  rest,  is  half  the  instantaneous 

velocity  at  the  end  of  the  second. 


KINEMATICS.  15 

15.  Second  Method.  —  Let  the  whole  time  t  be  divided 
into  a  very  large  number  n  of  equal  time-intervals  T. 

Then  nr=t. 

If  the  particle  starts  from  rest,  the  velocities  at  the  end 
of  the  several  small  time-intervals  are 

ar,  2ar,  3#r,     .     .     .     nar. 

If  these  time-intervals  are  sufficiently  small  the  velocity 
during  each  interval  may  be  assumed  constant.  Then  the 
spaces  described  in  the  successive  intervals  are 


The  entire  space  is  the  sum  of  these  elementary  spaces, 

or 

s  =  ar2  (1  +  2  +  3+     %     .     .     n^ 

To  find  the  sum  of  the  series  in  the  parenthesis  apply  the 
following  theorem  :  When  n  is  indefinitely  large  the  sum 
of  the  mth  powers  of  the  natural  numbers  1,  2,  3,  etc.,  to 

nm  ~f~* 

-     —  ,  where  of  course  m  is  the  exponent.1 

m+1 


n    s 


To  find  the  sum  of  the  series 


we  have  (Todhunter's  Algebra,  p.  404,  Hall  and  Knight's  Algebra,  p.  336). 

.  -  Cnm+  l  +  A,nm  +  *  A»m  ~  l  +  ^^  A2nm~*  +  etc.,  where 

L  L  *  O 


All  coefficients  with  even  subscripts  are  zero.    Hence 

n'»  +  1+i»wl+     *    nm-1-m^-1)^-2)n<n-8etc. 
«=  "  12  2- 3- 4 -30 

m  -\-  1 

If  we  suppose  the  series  cleared  of  fractions, 

as=n  +bnm  +  cnm          —etc., 

a  series  j>f  descending  powers  of  n. 
Omitting  constants,  s  is  of  the  form 

But  nm  +  l  =  nm  X   »,  and  »™  +  !  +  nm  =  nm  (n  +  1). 


16  MECHANICS. 

Therefore         1+2+3+.     .     .     n  =  ^, 

n~ 

since  wV  =  t2. 

PROBLEMS. 

1.  If  a  body  with  uniform  acceleration  acquire  a  velocity  of  10 
metres  a  second  in  moving  a  distance  of  25  metres  from  rest,  find  the- 
acceleration.     t/~   ~   JL*J.     •*  .       «-   «•   ^^r        <^t*-« .  •£  **•  fi**-  &*•• 

2.  In  what  time  will  a  body  moving  with,a  uniform  acceleration 
of  9  metres  a  second  traverse  250  metres  ?  »  (o  • 

3.  A  cannon  ball  has  a  muzzle  velocity  of  400  metres  a  second  ;  - 
the  length  of  the  gun  traversed  by  the  ball  is  3  metres.     Find  (a) 
the  acceleration  on  the  assumption  that  it  is  uniform;   (6)  the  time 
of  traversing  the  gun.       ^          .-*  > 

{  16.  The  Free  Fall  of  Bodies.  —  The  general  formulas 
connecting  space,  time,  velocity,  and  acceleration  in  uni- 
formly accelerated  motion  have  already  been  developed  in 
Articles  13  and  14.  If  we  consider  the  acceleration  of 
gravity  constant  at  any  place  on  the  earth's  surface,  and 
call  it  g,  then  we  obtain  the  following  formulas  by  substi- 
tuting for  a  the  particular  acceleration  in  question,  g. 
Hence 

V=fft 

(11) 


If  now  n  is  indefinitely  large,  then  unity  may  be  omitted  in  comparison  with  it, 
and  we  have  for  the  sum  of  the  first  two  terms  of  the  series 


Under  the  above  conditions,  therefore,  the  sum  of  the  first  two  terms  equals  the 
first  terra,  or  the  second  term  may  be  omitted  in  comparison  with  the  firs^.  The 
succeeding  terms  are  all  still  lower  powers  of  n,  and  they  are  therefore  negligible 
in  comparison  with  the  first  term.  It  follows  finally,  then,  that 


This  formula  will  be  found  to  be  very  useful  in  several  subsequent  topics. 


KINEMATICS.  17 

If  the  body  starts  with  an  initial  velocity  v0,  then 

v  =  v,  ±  gt       } 

v*=  v,2  ±  %*     V     .     .     .     .     (12) 

The  plus  sign  is  applicable -to  motion  downward  and  the 
minus  sign  to  projection  upward. 

PROBLEMS. 

1.  A  particle  has  a  uniform  acceleration  of  20  cms.  a  second,  and 
an  initial  velocity  of  30  cms.  a  second.     Find  (a)  the  velocity  after 
16  seconds  ;  (6)  the  time  required  to  travel  300  cms. ;  (c)  the  change 
in  velocity  in  traversing  that  distance. 

2.  A  body  is  projected  upward  with  any  velocity,  and  t  and  t'  de- 
note the  times  during  which  it  is  respectively  above  and  below  the 

middle  point  of  its  path  ;  find  the  value  of  j,> 

3.  A  body  dropped  from  the  top  of  a  tower  20  metres  high 
reached  the  bottom  of  a  well  within  the  tower  in  3  seconds  ;  find  the 
depth  of  the  well.     The  acceleration  of  gravity  is  9.8  metres  per 
second.  » 

17.  Projection  Upward.  —  If  motion  upward  is  consid- 
ered positive,  then  the  acceleration  is  negative  and  equa- 
tions (12)  must  be  applied  with  the  minus  sign.  The 
following  three  problems  may  then  be  solved : 

(CL)  To  find  the  time  of  ascent.  When  the  particle 
reaches  the  highest  point,  v  —  0.  Hence 

VQ  =  at     and  t  =  —  • 
9 

(5)  To  find  the  height  to  which  the  particle  will  ascend. 
Since  at  the  highest  point  v  =  0, 

g 

v02  =  2c/s     and  «  =   ~ . 
But  this  is  the  velocity  which  a  body  acquires  in  falling 


18  MECHANICS. 

freely  through  a  height  s.  Therefore,  neglecting  atmos- 
pheric resistance,  that  is,  allowing  the  particle  to  move 
freely,  it  will  return  to  the  point  of  projection  with  the  in- 
itial velocity  in  the  opposite  direction. 

(cs)  To  find  the  time  when  the  particle  will  be  at  a  given 
height.  Substitute  the  given  height  for  s  in  the  third 
equation.  This  gives  an  equation  of  the  second  degree 
with  two  roots.  The  physical  interpretation  is  that  the 
particle  will  be  at  the  given  elevation  once  while  ascend- 
ing and  again  when  descending. 

PROBLEMS. 

1.  A  body  is  projected  vertically  upward  with  an  initial  velocity 
of  250  metres  per  second.     Find  (a)  how  high  it  will  rise;   (6)  the 
time  of  ascent ;  (c)  when  it  will  be  350  metres  above  the  starting 
point. 

2.  A  body  thrown  vertically  upward  passes  a  point  10  metres 
from  the  starting  point  with  a  velocity  of  20  metres  a  second.     Find 
(a)  how  much  farther  it  will  go ;   (6)  its  initial  velocity. 

18.  Composition  of  Motions  (A.  and  B.,  16;  T.  andT.,  8 ; 
V.,  I,  48  ;  B.,  30).  —  Motions,  velocities,  and  accelerations 
are  directed  quantities,  and  they  are  capable  of  geometrical 
addition.  When  a  particle  has  several  motions  given  to 
it  at  the  same  time,  its  actual  motion  is  made  up  of  all  of 
them.  The  motions  are  then  said  to  be  compounded. 
The  actual  motion  is  the  resultant,  and  the  several  motions 
entering  into  it  are  called  the  components. 

When  there  are  two  uniform  rectilinear  motions  along 
the  same  straight  line,  their  resultant  is  their  sum  or 
difference,  according  as  they  are  similarly  or  oppositely 
directed. 


KINEMATICS. 


19 


O 


X' 


Let  us  now  consider  the  composition  of  two  uniform 
rectilinear  motions  taking 
place  along  different  right  O 
lines.  Let  the  particle  be 
subject  to  a  uniform  motion 
along  the  line  OX  (Fig.  5), 
while  at  the  same  time  this 
line  is  displaced  parallel 
to  itself,  so  that  the  point 
0  moves  uniformly  along 
OY. 

In  the  time  t  the  particle 
moving  along  OX  arrives 
at  the  point  P,  and 

OP  =  vt. 

But  OX  is  displaced  to  the  parallel  position  O'X',  and 

O0'=v't. 

The  particle  is  then  at  M,  the  line  O'M  being  taken  equal 
to  OP.     PMis  therefore  parallel  to  00'. 

In  another  time  t'  the  particle  will  arrive  at  P'  along 
OX,  and 

=  vt'. 


Fig.  5. 


In  the  same  time  the  line  will  have  arrived  at  the  position 
0"X"  and 


The  particle  will  then  be  at  M',  the  point  of  intersection 
of  the  parallel  to  OT  through  P'  with  0"X". 
But 

OP  =  «_=  00'       PM 
OP'       t'      00"~P'Mr 

The  triangles  POM,  P'  OM'  are  therefore  similar,  since 
the  angles  at  P  and  P',  contained  between  proportional 


20  MECHANICS. 

sides,  are  equal.  Hence  the  angles  of  the  two  triangles 
at  0  are  equal,  and  the  directions  OM  and  OM'  coin- 
cide. The  path  of  the  point  Mis  therefore  a  straight  line. 
The  motion  is,  moreover,  uniform.  From  the  same 
similar  triangles, 

OM      OP  _^ 

OM'  ~  OP'  ~  V 

The  spaces  described  along  the  diagonal  are  proportional 
to  the  times,  or  the  motion  is  uniform. 

We  see,  therefore,  that  if  the  two  motions  which  are 
applied  to  a  particle  at  the  same  time  are  represented  by 
the  adjacent  sides  of  a  parallelogram,  then  the  resultant 
motion  will  be  represented  by  the  diagonal  of  the  parallel- 
ogram drawn  through  the  intersection  of  these  two  sides. 

Accelerations  and  velocities  may  be  compounded  in  the 
same  manner  as  uniform  motions. 

19.    The  Triangle  of  Motions.  —  Let  two  motions  MA 

and  MB  be  given  to  the 
particle  at  M  at  the  same 
instant  (Fig.  6).  Construct 
upon  these  lines  as  adjacent 
sides  the  parallelogram 
AMBC  and  draw  through 
M  the  diagonal  MO.  MO 

represents  the  resultant  of  MA  and  MB  both  in  direction 
and  magnitude. 

Since  A  0  is  equal  and  parallel  to  MB,  the  three  sides  of 
the  triangle  MAO  may  represent  the  two  component 
motions  and  their  resultant.  That  is,  if  two  sides  of  a 
triangle  taken  in  order  represent  in  magnitude  and  direc- 
tion the  two  motions  applied  to  a  particle,  their  resultant 
will  be  represented  by  the  third  side  of  the  triangle. 


KINEMATICS.  21 

The  line  MO  is  called  the  geometrical  sum  of  the  two 
magnitudes  MA  and  MB.  Since  this  method  may  ob- 
viously be  extended  to  any  number  of  motions  by  means 
of  a  polygon  of  motions,  we  have  the  general  law  for  the 
composition  of  motions,  that  the  magnitude  of  the  resultant 
motion  is  the  geometrical  sum  of  the  magnitudes  of  the  com- 
ponent motions. 

20.    Given  Two  Motions  and  the  Angle  between  them, 
to  find  their  Resultant.  — 
Let  the  two  given  motions,  A2 - -^o 

P  and   Q)  be  represented 

by  the  lines  OA  and   OB 

respectively  (Fig.'  7)  ;  and 

let  A  OB  be  the  given  angle 

6.     Complete  the  parallel-  Fjg  7 

ogram.     It  is  /required  to 

find  the   value    of   the   resultant   R   represented   by   the 

diagonal  00. 

Since  AC  is  equal  to  OB,  it  may  equally  well  represent 
the  motion  Q.  From  plane  trigonometry, 


00*= 

Substituting  the  values  of  the  lines,  and  remembering  that 
the  angle  A  is  the  supplement  of  6>,  we  have 

R*  =  P2  +  #2  +  2P  •  Q  cos  6     .     .     .     (13) 

or  the  square  of  the  resultant  equals  the  sum  of  the 
squares  of  the  two  components  plus  twice  their  product 
into  the  cosine  of  the  included  angle. 

Three  particular  values  of  6  give  special  results. 

1.     When  0  =  0.     Then  cos  0  =  1,  and 

or  JB=P+  Q. 


22  MECHANICS. 

6  may  be  made  zero  by  rotating   OA  around  any  point  in 
it  as  0  or  A.     If  the  rotation  is  around  0  the  lines  OA 
and   OB  finally  coincide  when   6=0.     If   OA  is  rotated 
about  A,  then  when  0  =  0,  OA  is  parallel  to  OB. 
2.     When  6=  90°.     Then  cos  0=0,  and 


=  ,  or      = 

3.     When  6  =  180°.      Then   cos   8  =  -  1,   J22  =  P2  + 


Again  in  this  case  0  may  be  made  180°  by  rotating  OA 
counter  clockwise  around  either  0  or  A.  In  the  first 
case  the  motions  are  oppositely  directed  along  the  same 
straight  line,  and  in  the  second  case  they  are  oppositely 
directed  and  parallel.  This  result  is  of  importance  in 
compounding  parallel  forces  under  Kinetics. 

Finally,  when  P  equals  Q  with  any  angle  9 

H 

E  =  2P  cos  ~  , 
2 

for  the  component  of  either  motion  in  the  direction  of  the 
resultant,  which  now  lies  midway  between  P  and   Q,  is 

e 

Pcos-. 

PROBLEMS. 

1.  A  body  has  impressed  upon  it  two  velocities  of  25  and  20 
metres   a  second   at  an  angle  of  45  degrees.     Find  the  resultant 
velocity. 

2.  A  point  has  simultaneously  impressed  upon  it  three  velocities 
of  70,  60,  and  40  cms.  a  second.     The  angle  between  the  first  two  is 
60  degrees,  and  between  the  last  two  45  degrees.     Find  the  magni- 
tude of  the  resultant. 

21.  Resolution  in  any  Two  Rectangular  Directions.  - 
The  resolution  of  a  motion,  velocity,  or  acceleration  in  two 


KINEMATICS.  23 

rectangular  directions  is  of  more  frequent  occurrence 
than  any  other  case.  Let  AD  be 
a  velocity  v  (Fig.  8)  to  be  re- 
solved  into  rectangular  compo- 
nents. Let  AB  and  A  Q  be  the 
two  rectangular  directions.  Then 
the  two  components  required  are 
AB&ndAC. 

Let  a  and  j3  be  the  direction 

angles  which  v  makes  with  the  two  axes  respectively. 
Then 

AB  =  v  cos  a       .     .     .     .     o     .     (14) 

AC  =  v  cos  j3  =  v  sin  a. 

Hence  to  find  the  rectangular  component  of  a  velocity  in 
any  direction,  multiply  the  given  velocity  by  the  cosine  of 
the  direction  angle. 

22.  Extension  of  the  Parallelogram  of  Velocities.  — 
The  composition  and  resolution  of  motions  and  velocities 
have  thus  far  been  limited  to  those  varying  as  the  first 
power  of  the  time.  The  principle  of  the  parallelogram 
may  be  extended  to  motions  varying  as  any  function  of 
the  time,  provided  both  components  are  the  same  function. 
Thus  both  components  may  be  a  function  of  the  square  of 
the  time,  as  in  uniformly  accelerated  motion  ;  or  under  due 
limitations  they  may  vary  as  the  sine  or  cosine  of  an  angle 
which  is  proportional  to  the  time,  as  in  simple  harmonic 
motion.  The  resultant  will  then  be  represented  by  the 
diagonal  of  the  parallelogram,  and  will  be  the  same  func- 
tion of  the  time  as  the  components.  But  when  one  com- 
ponent is  one  function  of  the  time  and  the  other  another 
function,  then  the  extremity  of  the  diagonal  of  the  par- 
allelogram, constructed  on  the  two  lines  representing  the 


24  MECHANICS. 

motions  as  adjacent  sides,  will  be  the  position  of  the  par- 
ticle at  the  end  of  the  time  considered,  but  the  particle 
will  not  take  the  path  of  the  diagonal  to  arrive  at  the 
point. 

The  following  topic  will  illustrate  the  principle. 


23.  Path  of  a  Projectile.  — To  re- 
duce the  problem  to  its  simplest  form 
we  shall  assume  that  the  projectile 
meets  with  no  resistance  from  the 
atmosphere.  Let  v  be  the  velocity  of 
projection  in  the  direction  OX  (Fig. 
9).  This  motion,  which  is  uniform, 
must  be  compounded  with  a  uniformly 
accelerated  motion  in  a  vertical  di- 
rection OY. 

x  =  vt, 


Fig.  9. 


Then  in  time  £, 


Whence 


*•=*  -», 


(15) 


an  equation  of  the  form  xz  =  2py,  the  equation  of  a 
parabola. 

Or  we  may  reach  the  same  conclusion  geometrically  as 
folloAVS  : 

Let  OA  and  OB  be  distances  along  OX  proportional  to 
times  t  and  t'.  Let  00  and  OD  be  the  distances  de- 
scribed with  uniformly  accelerated  motion  along  OY  in. 
the  same  time-intervals.  Then 


Whence 


•  r  "" 

00  = 
OD 


OO 


KINEMATICS. 


25 


or  the  ordinates  are  proportional  to  the  squares  of  the 
abscissas.  This  is  a  property  characteristic  of  a  parabola. 
Equation  (15)  denotes  a  parabola  referred  to  a  tangent 
and  a  conjugate  diameter.  The  path  of  the  projectile  is 
then  a  parabola  tangent  to  the  direction  of  the  initial 
motion  and  having  its  axis  vertical. 

24.  To  find  the  Greatest  Elevation  and  the  Range 
(V.,  I,  201). — To  discuss  the  motion  of  a  projected  par- 
ticle it  is  more  conven- 
ient to  refer  it  to  rec- 
tangular axes.  Let  Ox 
(Fig.  10)  be  the  hori- 
zontal axis  passing 
through  the  point  of 
projection  0,  and  let  Oy 
be  the  vertical  through 
0.  Let  a  be  the  angle 
between  the  direction 
of  projection  and  the  horizon 
jection  v  may  be  resolved 
v  sin  a  vertical.  We  hav/e  th 


Fig.  10. 

The  velocity  of  pro- 
v  cos  a  horizontal,  and 
to  combine  a  uniform 


motion  v  cos  a  horizontally  and*  a  urj$£6rmly  retarded  mo- 
tion  along  the  vertical,  wji\\  the  initial\velocity  v  sin  a  and 
the  acceleration—  #  due  to  gravity.  The  equations  of  the 
motion  ar 

=  v  cosctf, 


Eliminating  t  gives  the  equation  of  the  path, 


=  x  tan  a  —  - 


This  is  the  equation  of  a  parabola  of  which  the  axis  is 
vertical. 


26  MECHANICS. 

The  summit  of  the  curve  is  easily  determined  if  we  con- 
sider that  at  that  point  the  vertical  component  of  the 
motion  is  zero. 

The  vertical  velocity  at  any  time  t  is 

v  sin  a  —  gt. 
Place  this  equal  to  zero  and 

f_v  sin  a 

9 

Substitute  this  value  of  t  in  the  general  expressions  for 
x  and  y  above  and 

,_  vz  sin  a  cos  a  _  v1  sin  2a        ,  _  v2  sin  2a 
g  2g       '    y         ~~Zg 

r™  , .  ,     .     ,,  , .  v'2  sin2  a     . 

The  particle  is  then  at  the  elevation  y'=  — ^ —  ;  it 

then  descends  along  a  branch  symmetrical  with  the  one 
traced  during  the  ascent.  The  range  a,  or  the  horizontal 
distance  to  the  point  where  it  will  again  touch  the  hori- 
zontal plane  through  the  point  of  projection,  is  therefore 
twice  the  value  of  x',  or 

_  v2  sin  2a 

9 
Since  v  is  a  constant,  the  greatest  range  will  correspond 

to  an  angle  of  elevation  of  45°.     The  range  is  then  — ,  or 

J 

double  the  height  which  the  projectile  would  attain  if 
fired  vertically  with  the  same  velocity  v. 

The  range  is  the  same  for  two  angles  of  elevation  equally 
distant  from  45°,  one  above  and  the  other  below. 

Practically  the  resistance  of  the  air,  which  we  have 
here  neglected,  will  greatly  modify  these  results,  especially 
with  large  velocities  of  projection. 


KINEMATICS. 


PROBLEMS. 

1.  A  piece  of  ordnance  under  proof  at  a  distance  of  75  metres 
from  a  wall  7  metres  high,  burst,  and  a  fragment  of  it,  originally  in 
contact  with  the  ground,  just  grazed  the  wall  and  fell  3  metres  be- 
yond it  on  the  opposite  side.     Find  how  high  it  rose  in  the  air. 

2.  If  a  body  be  projected  with  a  velocity  of  33  metres  a  second 
from  a  height  22  metres  above  the  ground  at  an  angle  of  elevation  of 
30°  with  the  horizontal,  find  when  and  where  it  will  strike  the  ground. 


25.  Motion  on  an  Inclined  Plane.  -  -  This  problem  is 
often  called  Galileo's  inclined  plane,  since  he  made  use  of  it 
for  the  purpose  of  diminishing  the  effective  component  of 
the  acceleration  of  gravity  in  determining  the  laws  of  falling 
bodies.  Let  the  material  particle  be  at  a  on  the  plane 
(Fig.  11),  the  angle  of 
elevation  of  which  is 
$.  The  acceleration 
g  to  which  the  particle 
is  subjected  is  in  a  ver- 
tical line,  while  the 
particle  is  constrained 
to  move  along  the 
plane.  Resolve  the  ac- 
celeration g  into  two  components,  one  normal  to  the  plane 
which  is  ineffective  in  producing  motion,  and  the  other 
parallel  to  the  plane,  the  effective  component.  The  latter 
is  g  sin  </>. 

Substitute  this  acceleration  for  a  in  equations  (8),  (9), 
(10),  and 


Fig    II 


v  =  g  sn 
s  =  &  sin 
v2—  2    sin 


(16) 


28 


MECHANICS. 


From  the  figure  sin  <f>  =  -;.     Hence  when  the  body  falls 
the  entire  length  of  the  plane 


The  velocity  attained  in  descending  the  entire  length 
of  the  plane  is  the  same  as  in  falling  down  the  vertical 
height  of  the  plane. 

With  the  same  conditions  the  second  equation  (16)  gives 


or 

.0* 

The  time  of    descending  a  plane    varies,  therefore,    as 
the  length  of  the  plane  if  the  height  remains  constant. 

PROBLEMS. 

1.  Find  the  gradient  of  a  railway  so  that  a  carriage  descending 
the  plane  by  its  own  weight  may  travel  400  metres  in  the   first 
minute ;  also  find  how  far  the  carnage  will  go  in  the  next  minute, 
friction  being  neglected. 

2.  A  body  slides  from  rest  down  a  sloping  roof  and  then  falls  to 

the  ground.     The  length  of  the  slope  is 
A  6  metres,  inclination  to  the  horizon  30°, 

and  the  height  of  its  lowest  point  from 
the  ground  13£  metres.  Find  the  dis- 
tance from  the  foot  of  the  wall  to  the 
point  where  the  body  strikes  the  ground. 

26.  The  Time  of  descent  down 
any  Chord  of  a  Vertical  Circle 
drawn  through  its  highest  point 
is  Constant.  —  Let  AC  (Fig.  12) 
be  any  chord  drawn  through  the 
highest  point  of  the  vertical  circle 
ABC.  AB  is  a  diameter.  Draw  CD  perpendicular  to  AB. 


KINEMATICS. 


29 


Then  the  acceleration  down  AC  is  gsin  ACD.     But  by 
similar  triangles 

the  angle  A  CD  =  the  angle  CBD. 

AC 

Therefore  g  sin  A  CD  =  g 

But  *  —  J$£2. 

AC 


Substituting 


Therefore 


-1(7= 


a  constant. 


The  time  of  descent  down  any  chord  through  A  is  there- 
fore a  constant  and  equal  to  the  time  of  falling  down  the 
vertical  diameter. 

27.     Uniform    Circular    Motion    (T.    and    T.,    9).  - 
Hitherto,  with  the  exception  of   the  parabolic  path  of  a 
projectile,  the  velocity  has  been  assumed  to  vary  in  mag- 
nitude only  ;  in  other  words,  the  acceleration  has  been  con- 
fined to  a  single  direction.    But  the  velocity  may  vary  also 
in   direction.     If    the  particle    has   a   uniform  rectilinear 
motion  the  acceleration  is  zero.     If  its  velocity  changes  in 
magnitude  without  any  change 
in  direction,  then  the  accelera- 
tion  is    positive    or   negative 
along  the  line  of  motion.     If, 
however,  the  direction  of  the 
motion  changes,  then  the  par- 
ticle has  an  acceleration  one 
component  of  which  is  at  right 
angles  to  its  path.     Thus  if  a 

particle  move  uniformly  along  AB  (Fig.  13),  while  at 
B  it  begins  to  describe  a  curved  path  from  B  to  (7,  and  from 
C  on  again  moves  uniformly  along  (7Z>,  then  between 
B  and  C  there  is  an  acceleration  normal  to  the  path. 


Fig.  13. 


80 


MECHANICS. 


Acceleration  should  therefore  be  extended  to  mean  rate  of 
change  of  velocity  in  any  direction  in  relation  to  the  path. 
In  uniform  circular  motion  the  speed  of  the  particle 
measured  along  the  circumference  is  constant,  while  the 
acceleration  is  also  constant,  and  is  directed  'toward  the 
centre.  It  is  required  to  find  the  value  of  this  centripetal 
acceleration.  .  . 

Let  v  be  the  velocity  along  the 
circumference.  Then  if  AD  (Fig. 
14)  be  traversed  in  the  interval  t, 

AD  =  vt. 

The  change  of  motion  from  AC, 
which  is  the  direction  of  the  path  at 
A,  to  AD  is  AE.    It  will  be  noted, 
however,  that  the  direction  of  the 
motion  at  D  is  not  AD,  but  the 
tangent  to  the    circle  through    D. 
The   change  in   direction  between 
A  and  D  is  therefore  twice  the  angle  CAD. 
Since  the  centripetal  acceleration  is  uniform, 


where  /is  the  required  acceleration. 

If  now  AD  is  an  indefinitely  short  arc,  then  the  triangles 
AED  and  ADB  are  similar,  and 

AE_AD 

AD 


AB' 


or 


Let  r  be  the  radius  of  the  circle.    Substituting  the  values 
of  AD,  AE,  and  AB,  and 


Whence 


/= 


(17) 


KINEMATICS.  31 

If  T  is  the  period  of  the  complete  revolution  of  the 
point,  then 

•  =  *£  -u*=4y. 

Substituting  this  value  of  vz  in  (17),  and 

/=^? (18) 

Since  a  radius  connecting  the  point  and  the  centre  de- 
scribes the  angle  2?r  in  the  period  T7,  the  ratio  -=  =  a)  is 

called  the  angular  velocity.     Equation  (18)  'may  therefore 
be  written 

/=•* (19) 

PROBLEMS. 

1.  A  mass  of  1  gm.  moves  uniformly  round  a  circle  40  cms.  in 
diameter  at  the   rate  of  24  revolutions   a   minute.      Compute  the 
acceleration  toward  the  centre. 

2.  Two  equal  masses,  A  and  B,  are  connected  by  a  string.     The 
mass  A  describes  a  circle  of  radius  1  metre  with  uniform  speed  on 
the  surface  of  a  smooth  horizontal  table,  while  the  other  mass  B  is 
suspended  against  gravity  by  the   string,  which  passes  through  a 
small,  smooth  hole  at  the  centre  of  the  table.     Find  the  speed  of  A, 
assuming  the  acceleration  of  gravity  to  be  980  cms.  per  second. 

28.  Simple  Harmonic  Motion  (T.  and  T.,  19  ;  D.,  79  ; 
A.  and  B.,  18;  V.,  II,  260).  —  Simple  harmonic  motion 
is  one  of  the  most  important  motions  which  we  have  to 
consider,  on  account  of  its  frequent  use  in  sound,  light, 
and  alternating  currents  of  electricity.  While  adhering, 
therefore,  to  an  elementary  method  of  treatment,  we  shall, 
nevertheless,  discuss  it  somewhat  in  detail. 

Simple  harmonic  motion  is  the  apparent  motion  of  a 
point,  describing  uniform  circular  motion,  when  viewed 


MECHANICS. 


from  a  great  distance  in  the  plane  of  the  circle.  It  is  the 
component  of  the  uniform  circular  motion  at  right  angles  to 
the  line  of  sight ;  or  it  is  the  motion  of  the  intersection  of 
a  diameter  of  the  circle  and  a  perpendicular  from  the  mov- 
ing point  on  this  diameter.  The  simple  harmonic  motion 
is  along  a  straight  line  with  a  maximum  velocity  at  its 
middle  point  and  zero  at  the  extremities. 

The  satellites  of  Jupiter  revolve  in  orbits  which  coin- 
cide very  nearly  with  the  plane  of  the  ecliptic,  or  in  a 
plane  which  passes  nearly  through  the  earth.  Hence  they 

appear  to  travel  slowly  back- 
ward and  forward  in  nearly 
straight  lines  with  simple  har- 
monic motion. 

If  the  circle  QAR  (Fig.  15) 
represents  the  circle  in  which 
the  point  is  moving,  QR  its  ap- 
parent linear  path,  and  A,  B, 
(7,  D,  etc.,  some  of  its  successive 
positions,  we  may  define  its  ap- 
parent motion  by  drawing  per- 
pendiculars from  A,  B,  C,  etc.,  and  finding  the  points 
a,  ft,  c,  etc.,  to  which  the  several  positions  of  the  point  in 
the  circle  correspond.  If  the  points  on  the  circle  are  laid 
off  at  equal  distances,  then  their  corresponding  projections 
on  the  diameter  QR  will  not  be  equidistant ;  the  distances 
Qa,  ab,  be,  etc.,  will,  however,  represent  the  spaces  appar- 
ently traversed  in  equal  intervals -of  time. 

The  motion  along  the  diameter  is  not  uniform,  but  is 
oscillatory,  and  the  acceleration  is  directed  toward  the 
middle  point  of  the  diameter. 

The  circle  is  called  the  auxiliary  circle  or  circle  of  refer- 
ence, and  its  radius  is  the  amplitude  of  the  simple  harmonic 
motion. 


Fig.   15. 


KINEMATICS. 


33 


The  period  of  the  motion  is  the  time  of  a  complete  revo- 
lution of  the  point  around  the  circle  of  reference. 

Motion  from  left  to  right  is  positive,  and  from  right  to 
left  negative.  Displacement  to  the  right  of  the  middle 
point  is  positive,  and  to  the  left  negative. 

The  phase  is  the  fraction  of  a  whole  period  which  has 
elapsed  since  the  particle  last  passed  through  the  middle 
of  its  range  in  the  positive  direction. 

\Vhen  the  particle  is  at  Q  it  is  said  to  be  at  its  greatest 
positive  elongation. 

The  epoch  is  the  interval  which  has  elapsed  from  the 
point  of  reckoning  time  till  the  particle  arrives  at  zero 
elongation  going  in  the  positive  direction.  In  angular 
measure  it  is  the  angle  described  on  the  circle  of  reference 
during  the  time-interval  defined  as  the  epoch. 


'29.  Acceleration  in  Simple  Harmonic  Motion  propor- 
tional to  Displacement.  —  Let  the  particle  be  at  the  point 
B  in  the  circle  of  reference  (Fig. 
16).  The  displacement  is  the  line 
00.  The  acceleration  toward 

A      2 

the  centre  is  — -r,  since  the  par- 
ticle is  moving  uniformly  aroun'd 
the  circle.  The  problem  is  to 
resolve  this  centripetal  accelera- 
tion into  two  rectangular  com- 
ponents, parallel  to  OX  and  OY 
respectively. 

Denote  the  two  component  accelerations  by  /x  and  fr 
Then,  since  the  component  in  any  direction  is  found  by 
multiplying  by  the  cosine  of  the  direction  angle, 


Fig.   16. 


34 


MECHANICS. 


/*=    - 


«ja 


X  sin  0. 


Call  the  two  displacements  along  the  two  axes  #  and 
respectively  ;  then 


cos  6  =    ,  sin  6  —  3-. 
r  r 


4-7T 


4?r2 


_  _ 

2T-  =  2 

/.  4-7T2 


Therefore 


The  first  component  varies  as  x  and  the  second  one  as 
y  ;  or  the  two  components  of  the  centripetal  acceleration 
are  proportional  to  the  displacements  of  the  points  from  the 
two  rectangular  diameters. 

It  is  to  be  observed  that  when  one  of  these  components 
is  a  maximum  the  other  is  a  minimum,  or  they  differ  in 
phase  by  a  quarter  of  a  period. 

Now,  since  the  only  acceleration  in  uniform  circular 
motion  is  directed  toward  the  centre  of  the  circle  and  is 
constant  in  value,  it  follows  that  uniform  circular  motion 

may  be  resolved  into  two  sim- 
ple harmonic  motions  at  right 
angles  to  each  other,  of  the  same 
period  and  amplitude,  and  dif- 
fering in  phase  by  a  quarter  of 
a  period. 

3O.     The   Velocity    at   any 
Point    (A.  and  B.,  19).  —  To 
find  the  velocity  of  the  simple 
harmonic  motion    at  any  point 
C  (Fig.    17),    resolve    the    uniform    velocity    V  in    the 


KINEMATICS.  35 

auxiliary  circle  into  two  components  parallel  to  the  axes 
of  X  and   Y. 

Represent  the  velocity  at  B  in  the  circle  by  BE.  Com- 
plete the  right  triangle  BEF.  Then,  since  BE  is  a 
tangent,  the  angle  E  equals  6.  Hence  BF,  the  compo- 
nent parallel  to  the  X  axis,  is 

Fx=    -  Fsin0. 
Also  Vy=  Fcos0. 


. 

Therefore  Fx  =  -  2?rr  sin  0, 


When  0=  7r,  sin  0  =  1,  and  cos  0  =  0. 

4 

When  one  component  of  the  velocity  is  a  maximum  the 
other  is  a  minimum.  The  maximum  velocity  in  simple 
harmonic  motion  is  the  same  as  the  uniform  velocity  in  the 
circle  of  reference. 

It  is  important  to  observe  that  while  the  acceleration 
along  the  X  axis  is  proportional  to  cos  0,  the  velocity  is 
proportional  to  sin  0.  When  the  point  is  at  the  middle 
of  its  range  or  course  the  acceleration  is  zero,  but  the 
velocity  is  a  maximum;  on  the  other  hand,  at  the  greatest 
elongation  the  acceleration  is  a  maximum,  and  the  velocity 
is  zero.  Starting  at  the  extreme  positive  elongation  at  Jf, 
the  acceleration  decreases  continuously  toward  the  centre, 
while  the  velocity  increases  ;  but  it  increases  at  a  contin- 
uously decreasing  rate. 

If  the  earth  were  spherical  and  of  uniform  density,  and 


36  M  EC  II A  NICU. 

if  a  hole  could  be  drilled  through  its  centre  to  the  opposite 
side,  then,  neglecting  resistance  of  the  air  and  the  rotation 
on  the  earth's  axis,  a  heavy  body  dropped  into  the  hole 
would  descend  with  a  diminishing  acceleration  and  a  con- 
stantly increasing  velocity  till  it  reached  the  centre ;  the 
acceleration  would  then  become  negative,  and  the  velocity 
of  the  body  would  decrease,  till  at  the  opposite  surface  it 
would  again  come  to  rest,  and  would  retrace  its  course, 
executing  simple  harmonic  oscillations.  Such  is  the  motion 
of  a  point  on  a  tuning  fork  or  a  pianoforte  wire,  aiid  such 
is  the  motion  of  individual  particles  of  air  through  Avhich 
a  simple  fundamental  tone  is  passing.  The  variations  of 
electric  pressure  in  the  circuit  of  an  alternating  current 
dynamo  follow  approximately  the  same  law  of  change. 

PROBLEM. 

A  horizontal  shelf  moves  vertically  with  simple  harmonic  motion, 
the  complete  period  being  one  second.  Find  the  greatest  amplitude 
it  can  have  in  centimetres  so  that  objects  resting  on  it  may  remain  in 
contact  with  it  at  its  highest  point,  assuming  g  equal  to  980. 

31.    To  find  the  Resultant  of  Two  Simple  Harmonic 
Motions  of  the  same  Period  and  in  One  Line  (T.  and  T., 
21).  —  Let  a  and  b  be  two  points  executing  simple  harmonic 
motions  of  the  same  period  in  the  line  OY  (Fig.  18).    Let 
A     and    B   be    the    corresponding    uni- 
formly moving   points    in  the    circles    of 
reference. 

On  OA  and  OB  describe  a  parallelo- 
gram and  draw  Aa,  Bb,  and  Ca  perpen- 
dicular to  OY.  Then  the  angle  AOB  is 
the  difference  in  phase  between  the  two 
harmonic  motions  to  be  compounded  ;  and 
since  they  have  the  same  period,  this  phase-difference  has 


KINEMATICS.  37 

a  fixed  value,  and  the  diagonal  of  the  parallelogram  remains 
of  constant  length,  and  makes  constant  angles  with  OA 
and  OB.  The  point  O  therefore  moves  uniformly  around  a 
circle  of  radius  00,  and  the  point  c  executes  simple  har- 
monic motion  of  the  same  period  as  a  and  b. 

Moreover,  be  equals  Oa,  since  they  are  the  projections 
on  OF  of  the  equal  parallel  lines  BO  and  OA.  Hence 

Oc  =  Ob  +  be  =  Ob  +  Oa. 

But  jOa,  Ob,  and  Oc  are  the  instantaneous  values  of  the 
displacements  of  A,  B,  and  C  at  any  time  t.  The  motion 
of  c  is  therefore  the  resultant  of  the  motions  of  a  and  b. 
It  is  a  simple  harmonic  motion  of  the  same  period  as  that 
of  the  component  motions.  The  resulting  amplitude  is 
the  diagonal  of  the  parallelogram  constructed  on  the  two 
component  amplitudes,  and  making  an  angle  with  each 
other  equal  to  their  fixed  difference  of  p'hase. 

Let  a  parallelogram  of  cardboard,  OACB,  be  cut  out 
and  be  pivoted  at  0  so  as  to  turn  freely  counter  clockwise. 
As  the  cardboard  moves  uniformly  around,  the  magnitudes 
of  the  projections  of  the  two  sides  OA  and  OB  and  the 
diagonal  OC  fluctuate  according  to  a  simple  periodic  law. 

The  resultant  of  the  simple  periodic  motions,  of  which 
OA  and  OB  are  the  amplitudes,  and  which  have  a  fixed 
phase-difference  equal  to  the  angle  AOB,  is  the  simple 
harmonic  motion  of  which  OC  is  the  amplitude. 

The  proposition  is  independent  of  the  angle  represent- 
ing the  fixed  difference  of  phase,  and  therefore  holds  good 
when  this  difference  is  zero.  In  this  last  case  not  only  is 
the  resulting  displacement  equal  to  the  algebraic  sum  of 
the  component  displacements,  but  the  resulting  amplitude 
equals  the  algebraic  sum  of  the  amplitudes  of  the  com- 
ponent motions. 


38 


MECHANICS. 


If  the  two  amplitudes  of  the  component  motions  are 
equal  to  each  other,  then  the  resulting  amplitude  equals 
twice  the  amplitude  of  either  multiplied  hy  the  cosine  of 
half  the  difference  of  phase  (20). 

When  this  difference  of  phase  is  half  a  period  or  TT,  then 
the  resultant  is  zero. 

When  the  two  periods  are  nearly  but  not  quite  equal, 
while  the  amplitudes  are  still  supposed  equal,  then  the 
resulting  amplitude  passes  slowly  from  twice  that  of  either 
to  zero  and  back  again,  in  a  time  equal  to  that  required  for 
the  faster  to  gain  a  complete  oscillation  on  the  slower. 

This  conclusion  has  important  applications  in  sound 
and  physical  optics. 

32.  Composition  of  Two  Simultaneous  Circular  Mo- 
tions (D.,  99).  —  Let  the  two  uniform  circular  motions  be 

in  opposite  di- 
rections around 
circles  of  equal 
radii  (Fig.  19), 
and  let  them  be 
of  the  same  pe- 
riod. 

Resolve  the 
two  circular  mo- 
tions into  simple 
harmonic  com- 
ponents parallel  to  AB  and  CD.  If  now  the  two  circular 
motions  are  conceived  to  be  applied  to  the  same  particle, 
then  since  the  harmonic  components  of  the  two  circular 
motions  parallel  to  CD  are  equal  and  oppositely  directed, 
their  resultant  is  zero.  But  those  parallel  to  AB  are  in 
the  same  direction;  and  since  they  are  simple  harmonic 


KINEMATICS. 


39 


motions  and  of  the  same  period  and  phase,  their  resultant 
is  a  simple  harmonic  motion  of  the  same  period  as  the 
circular  motions,  and  the  amplitude  is  twice  the  common 
radius  of  the  circles. 

The  resultant  simple  harmonic  motion  is  in  the  plane  of 
symmetry  with  respect  to  the  two  circles.  The  com- 
ponents CD  differ  in  phase  by  half  a  period  when  referred 
to  the  plane  of  symmetry,  while  the  components  AB  are 
in  the  same  phase  with  respect  to  the  plane  of  symmetry. 
When  the  two  circular  motions  have  the  same  period,  a 
plane  of  symmetry  can  always  be  found,  and  the  resultant 
simple  harmonic 

motion   will   be  / 

parallel  to   this  / 

plane  (Fig.  20). 

If  the  periods 
of  the  two  cir- 
cular motions 
are  not  precisely 
equal,  then  one 
of  the  circular 
motions  com- 
pletes a  revolu- 
tion before  the 
other,  and  the 
plane  of  symmetry  revolves  in  the  direction  of  the  circular 
motion  of  shorter  period.  It  makes  one  revolution  while 
one  circular  motion  gains  a  complete  revolution  on  the 
other.  This  principle  has  an  important  application  in 
the  rotation  of  the  plane  of  polarization  of  plane  polarized 
light  by  such  substances  as  quartz,  solutions  of  sugar,  and 
by  the  action  of  a  magnetic  field. 

The  converse  of  the  above  proposition  is  obviously  true, 


Fig.  20. 


40 


MECHANICS. 


viz.,  that  any  simple  harmonic  motion  may  be  resolved 
into  two  uniform  circular  motions  in  opposite  directions,  of 
the  same  period  as  the  simple  harmonic  motion,  and  with 
radii  equal  to  half  its  amplitude. 

33.    General  Equations  of  Simple  Harmonic  Motion. - 

Let  x  be  the  displacement  par- 
allel to  AB  (Fig.  21).  It  is 
required  to  find  the  value  of 
this  displacement.  Let  the 
point  in  the  auxiliary  circle  be 
at  P. 

Then  x  equals  OS,  and 

x  =  a  cos  B  OP, 
or  x  =  a  cos  ^>. 

But  the  definition  of  phase 
usually  adopted  renders  it  nec- 
essary to  express  x  as  a  sine  function  of  an  angle. 


Then 


x  =  a  sin 


Put  6  for  <f>  +  £,  or  the  angle  DOP;  then 
J 

x  =  a  sin  6. 

Let  t  be  the  time  of  describing  the  angle  0  and  T  the 
period  of  a  complete  revolution,  or  of  a  complete  simple 
harmonic  oscillation.  Then  since  the  circular  motion  is 
uniform, 

i  -  t 

27T  ~  T  ' 


.and- 


KIN  EN  A  TIC8.  41 

We  may  therefore  write 

27T 

x  =  a  sin  -m  t, 

or  x  =  a  sin  ait. 

But  it  is  often  necessary  to  reckon  time  or  the  angle  9 
from  some  otfrer  point  than  D,  which  corresponds  to  a 
displacement  of  zero.  This  is  for  the  purpose  of  expressing 
difference  of  phase  when  two  or  more  simple  harmonic 
motions  are  compounded.  Let  time  be  reckoned  from  the 
fixed  radius  OE ;  the  angle  EOD  is  called  the  epoch  e. 
The  angle  6  then  becomes  EOP.  Hence 

x  =  a  sin  (0  —  e), 
or  x  =  a  sin  (  ^-  t  — 


Since  the  number  of  complete  oscillations  per  second  n 
is  the  reciprocal  of  the  period  T,  or  n  =  •=,  we  have 

x  =  a  sin  (fyirnt  —  e). 
Similarly  we  may  write 

v  27T     .  /27T, 

^a  cos  f-^ 

47T2  /2-7T  .  \ 

yTa   S111   (^  )' 

Simple  harmonic  motion  is  therefore  an  oscillation  in 
which  the  acceleration  is  proportional  to  the  sine  of  an 
angle  varying  directly  as  the  time.  Since  the  sine  of  an 
angle  has  regularly  recurring  values,  simple  harmonic 
motion  is  a  simple  periodic  function  of  the  time  ;  that  is, 
the  displacement,  acceleration,  and  velocity  are  simple 
periodic  functions. 


42  MECHANICS. 


CHAPTER    III. 


KINETICS. 

34.  Definition  of  Kinetics  (T.  and  T.,  52  ;  B.,  66).  - 
Hitherto  motion  has  been  considered  in  the  abstract,  inde- 
pendently of  the  idea  of  matter  and  of  the  forces  acting 
on  it.     But  in   Kinetics  the  forces  producing  the  motion 
and  the  quantity  of  matter  moved  must  be  taken  into  ac- 
count.    For  the   velocity  of  a  body  depends  not  only  on 
the  magnitude  of   the  forces  acting,  but   also    upon   the 
quantity  of  matter  set  in  motion.     Kinetics  treats  of  the 
action  of  forces  in  producing  the  motion  of  definite  quan- 
tities of  matter.  . 

35.  Mass,  Volume,  Density.  —  Mass  is    the  quantity 
of  matter  contained  in  a  body.     It  is  expressed  numeri- 
cally in  terms  of  the  unit  of  mass,  the  gramme. 

Volume  is  the  space  which  a  body  occupies,  expressed 
in  cubic  centimetres  as  the  unit  of  volume. 

Density  is  the  mass  of  matter  contained  in  unit  volume. 
In  the  C.G.S.  system  it  is  the  number  of  grammes  of 
matter  in  a  cubic  centimetre.  The  mean  density  of  a 
body  may  therefore  be  found  by  dividing  its  mass  in 
grammes  by  its  volume  in  cubic  centimetres.  The  relation 
between  density,  mass,  and  volume  may  be  written 

•    j        >»  ur 

a=        and  m  =  a  V . 


KINETICS.  43 

It  is  often  convenient  to  employ  the  volume  containing 
unit  mass.  Call  it  s.  Then  d  and  s  are  reciprocals  of  each 

other,  or  d  =  —  . 

8 

36.  Momentum The  quantity  of  motion  of  a  rigid 

body  moving  without  rotation  is  considered  to  be  made  up 
of  its   mass  arid  its  velocity  conjointly.     It  is  called  mo- 
mentum. 

Momentum  =  mv. 

The  whole  motion  of  a  body  is  the  sum  of  the  motions 
of  its  several  parts.  If  the  mass  be  doubled  without 
changing  the  velocity,  or  if  the-  velocity  be  doubled  with- 
out changing  the  mass,  then  in  both  cases  the  quantity  of 
motion  is  doubled. 

Since  in  uniform  motion  v  —  — ,  we  have  mv  =  — •. 

L  •  t/ 

Speed  is  the  rate  of  linear  displacement ;  momentum  is 
the  rate  of  mass  displacement. 

« 

37.  Force.  —  Our  conception  of  force  is  doubtless  de- 
rived from   muscular  action.     But  great    caution  should 
be  observed  in  transferring  to  the  objective  physical  world 
any  concepts  derived  from  our  sensations. 

Force  is  said  to  act  on  a  body  when  any  change  occurs 
in  its  state  of  rest  or  motion.  Force  is  known  by  the 
change  it  produces  in  the  motion  of  a  body. 

The  intensity  of  a  force  is  measured  by  the  acceleration 
which  it  imparts  to  unit  mass. 

The  total  magnitude  of  a  force  is  the  product  of  mass  and 

acceleration,  or  F  =  m  V-~    ~~  =  ma.     This  acceleration  may 

t 

be  a  change  in  either  the  magnitude  or  the  direction  of  the 
velocity.     Weight  is  the  total  force  of  gravity,  or  W=  mg. 


44  MECHANICS. 

rni  .  v — VQ       mv — mVn  p  i  •  i   .,  . 

1  he  expression  m = -,  from  which  it  is  seen 

that  total  force  equals  the  time-rate  of  change  of  momen- 
tum. This  is  sometimes  called  the  acceleration  of  mo- 
mentum. 

The  rate  of  change  of  momentum  of  a  falling  body  is 
constant,  and  in  the  vertical  direction.  But  the  rate 
of  change  of  the  acceleration  of  a  mass  M,  describing  a 

circle  of  radius  r,  with  uniform  velocity  v,  is  M  —  ,  and  is 

directed  toward  the  centre  of  the  circle  ;  that  is  to  say,  it 
depends  upon  a  change  of  direction  of  the  motion,  not  a 
change  of  speed  (T.  and  T.,  53). 

An  impulsive  force  acts  for  a  very  short  period  only, 
while  a  continued  force  acts  for  a  longer  or  sensible  period. 
The  only  difference  is  one  of  time.  The  time  element  is 
essential  to  produce  any  effect.  When  a  hammer  strikes 
an  anvil  it  remains  in  contact  with  it  for  a  measurable 
interval  of  time*  So  also  when  a  bat  strikes  a  ball  the 
two  remain  in  contact  long  enough  for  the  ball  to  be 
brought  to  rest  and  to  acquire  motion  in  the  opposite 
direction.  Any  force,  however  great,  can  produce  only 
zero  effect  in  zero  time.  Hence  in  estimating  the  action 
of  a  force  the  time  element  is  of  equal  importance  with  the 
magnitude  of  the  force.  The  word  "  impulse  "  takes  both 
into  account. 

Impulse  is  the  product  of  the  force  and  the  time  during 
which  it  acts. 

The  practical  unit  of  force  is  the  dyne.  It  is  that  force 
which  acting  on  a  mass  of  one  gramme  for  one  second 
gives  to  it  a  velocity  of  one  centimetre  per  second.  It 
produces  unit  acceleration  of  unit  mass. 

The  intensity  of  a  force  is  measured  by  the  acceleration 


KINETICS.  45 

it  produces.  (  Hence  we  may  substitute  force  for  accelera- 
tion in  the  preceding  propositions. 


PROBLEMS. 

What  is  the  acceleration  when  a  force  of  36  dynes  acts  OIL  a 
>f  4  gins.  ?     How  far  will  the  mass  move  in  10  seconds  ?       /  * 
A  force  of  60  dynes  acts  on  a  body  foY  one  minute  and  imparts     /_- 
to  it  a  velocity  of  900  cms.  a  second.    What  is  the  mass  of  the  body  ?  ' 

3.  A  mass  of  500  gins,  is  whirled  around  at  the  end  of  a  string 
20  cms.  long  3  times  a  second.     What  is  the  tension  of  the  string, 
neglecting  gravity? 

4.  Forces   of  20,  30,  40  dynes  act  on  a  mass  of  60  gins. ;  the 
angle  between  the  first  two  is  45°,  and  that  between  the  last  two  60°. 
Find  the  space  passed  over  in  10  seconds. 

5.  An  engine  winds  a  cage  weighing  3,000  kilos,  up  a  shaft  at  a 
uniform  speed  of  10  metres  a  second;  what  is  the  tension  in  the 
rope  ?     What,  if  the  cage  move  with  a  uniform  acceleration  of  10 
metres  a  second? 

6.  An  elevator  starts  to  descend  with  an  acceleration  of  3  metres 
a  second.     Find  the  pressure  on  its  floor  of  a  man  weighing  75  kilos. 
What  would  be  his  weight  with  respect  to  the  elevator  if  it  started  to 
ascend  with  the  same  acceleration  ? 

38.    Newton's  Laws  of  Motion  (T.  and  T.,  64  and  65). 

-  Jfhe  relations  of  motions  and  changes  of  motion  to  the 
forces  producing  them  are  expressed  in  Newton's  three 
laws.  These  are  to  be  regarded  as  physical  axioms,  which 
are  not  susceptible  of  rigorous  demonstration.  They  are 
axiomatic  to  those  only  who  have  sufficient  knowledge  of 
physical  phenomena  to  enable  them  to  interpret  their  rela- 
tions. The  laws  of  motion  must  be  considered  as  resting 
on  convictions  drawn  from  observation  and  experiment. 

Law  I.  —  Every  body  continues  in  its  state  of  rest  or  of 
uniform  motion  in  a  straight  line,  except  in  so  far  as  it 
may  be  compelled  by  impressed  force  to  change  that  state. 

Law  II.  —  Change  of  motion  is  proportional  to  the  im- 


46  MECHANICS. 

pressed  force,  and  takes  place  in  the  direction  of  the 
straight  line  in  which  the  force  acts. 

Law  III.  —  To  every  action  there  is  always  an  equal  and 
contrary  reaction ;  or  the  mutual  actions  of  two  bodies  are 
always  equal  and  oppositely  directed. 

Newton  defined  the  total  motion  of  a  body  by  the  term 
momentum.  By  "  change  of  motion  "  we  should,  therefore,* 
invariably  understand  change  of  momentum. 

Further,  since  the  effect  produced  by  a  force  depends 
upon  the  time  during  which  it  acts  as  well  as  upon  the 
magnitude  of  the  force,  "  impressed  force  "  should  always 
be  interpreted  as  meaning  impulse. 

39.  Discussion  of  the  First  Law  of  Motion  (T.  and  T., 
65-69;  D.,  5;  B.,  7O). — The  first  law  asserts  that  uni- 
form motion  in  a  straight  line  is  as  much  the  natural  con- 
dition of  a  body  as  rest.  The  ideas  embodied  in  the  law 
are  completely  at  variance  with  those  of  the  ancient  phi- 
losophers who  asserted,  on  purely  metaphysical  grounds 
apparently,  that  circular  motion  is  the  only  perfect  one. 

The  term  rest  must  be  taken  to  mean  rest  with  respect 
to  other  contiguous  or  related  bodies.  Absolute  rest,  no 
less  than  absolute  motion,  is  unknown  in  nature.  While, 
therefore,  a  body  is  at  rest  with  respect  to  one  body  or  to  a 
system  of  bodies,  it  is  at  the  same  time  in  motion  when 
another  body  or  system  of  bodies  is  considered. 

When  no  force  acts  011  a  body  it  persists  in  its  state  of 
rest  or  motion  in  relation  to  other  bodies.  Thus  when  a 
ball  is  fired  from  a  cannon  it  continues  to  move,  not 
because  the  force  of  the  explosion  follows  it  and  keeps 
moving,  but  because  it  meets  with  nothing  capable  of  stop- 
ping it.  It  keeps  moving  unless  something  stops  it.  The 
question  of  the  energy  that  it  conveys  from  the  source  of 
motion  is  reserved  for  discussion  in  a  later  section. 


KINETICS. 


47 


Matter  has  no  power  in  itself  to  change  its  condition  of 
rest  or  of  motion.  Further,  it  offers  resistance  to  any  such 
change  in  proportion  to  thQjnass  which  it  contains.  This 
two-phased  property  of  matter  is  expressed  by  the  term 
Inertia,  and  the  first  law  of  motion  is  often  described  as 
the  law  of  inertia. 

4O.  Inertia  illustrated.  —  When  a  stream  of  water  in 
a  pipe  is  suddenly  stopped  by  the  closing  of  a  valve,  the 
shock  which  follows  is  due  to  the  inertia  of  the  mass  of 
water  which  tends  to  keep  it  moving.  This  effect  is 
utilized  in  the  hydraulic  ram.  The  shock  due  to  the  auto- 
matic closing  of  one  valve  opens 
another  one  leading  into  an  air 
chamber.  This  latter  valve  opens 
under  greater  pressure  than  that 
due  to  the  head  of  water  in  the 
pipe  leading  from  the  source. 
Hence  a  part  of  the  water  may 
be  lifted  to  a  higher  level  than 
the  source  of  supply. 

In  earthquake  shocks  accom- 
panied by  gyratory  movements  of 
the  ground,  heavy  chimneys  and 
isolated  columns  are  sometimes 
left  twisted  around  on  their  foun- 
dations. The  inertia  of  the  mass 
of  the  chimney  or  column  causes 
it  to  remain  fixed  while  the  earth 
suddenly  turns  under  it.  A  slower 
reverse  gyration  of  the  ground  turns  the  column  around 
with  it. 

Suspend  by  a  string  a  heavy  weight  A  (Fig.  22),  and 
attach  below  by  a  piece  of  the  same  string  the  bar  B.  A 


48  MECHANICS. 

steady  pull  downward  on  B  will  cause  the  string  to  break 
above  A,  because  the  tension  of  the  upper  string  is  equal 
to  the  weight  of  A  in  addition  to  the  pull  on  B. 

If,  however,  a  sudden  pull  be  applied  to  B  the  string 
will  invariably  break  below  A.  The  inertia  of  the  weight 
A  is  so  great  that  the  lower  string  breaks  before  the 
stress  reaches  the  upper  one. 

41.  Discussion  of  the  Second  Law  (L.,  53).  —  The 
first  law  defines  the  condition  under  which  a  change  of 
momentum  takes  place.  The  second  law  shows  us  first 
how  a  force  may  be  measure'd.  "  Change  of  motion  is 
proportional  to  the  impressed  force."  Maxwell  has  re- 
stated this  law  in  the  following  language  :  "  The  change  of 
momentum  of  a  body  is  numerically  equal  to  the  impulse 
which  produces  it,  and  is  in  the  same  direction"  The  sub- 
stitution of  the  word  "equal  "  for  "proportional  "  depends 
upon  the  definition  of  the  unit  of  measurement.  Impulse 
may  therefore  be  placed  equal  to  the  change  of  momentum 
produced  by  it,  or 

Ft  =  mv  —  mvQ. 


TTTI  rr 

Whence  F  = 


6 

This  is  the  same  expression  for  force  as  that  found  in 
Art.  37.  When  there  is  no  change  of  momentum  there  is, 
therefore,  no  force.  This  is  only  the  substance  of  the  first 
law  in  another  form.  The  first  law  is,  therefore,  involved 
in  the  second. 

The  second  law  shows  further  that  the  directijp.  of  the 
change  of  momentum  always  coincides  with  the  direction 
of  the  impulse.  This  means  that  a  force  always  produces 
its  full  effect  on  the  motion  of  a  body  whatever  its  previous 
condition  of  motion  or  direction  of  motion.  It  implies 


KINETICS.  49 

also  that  if  two  or  more  forces  act  together  on  a  body  each 
produces  its  own  change  of  momentum  independently  of 
the  others. 

We  may  therefore  employ  the  same  principles  to  com- 
pound forces  that  we  have  already  employed  for  compound- 
ing motions.  Forces  may  be  compounded  into  a  resultant 
or  resolved  into  components  by  means  of  the  parallelogram 
or  triangle  of  forces  in  the  same  manner  as  motions.  The 
resultant  of  any  number  of  concurring  forces  is  to  be  found 
by  the  same  geometrical  process  as  the  resultant  of  any 
number  of  simultaneous  velocities  (T.  and  T.,  67). 

42.  Discussion  of  the  Third  Law  (M.  and  M.,  77).  — 
The  third  law  expresses  the  fact  that  all  action  of  force  is 
of  a  dual  character.  All  action  between  bodies  or  the 
parts  of  a  system  of  bodies  is  of  the  nature  of  a  stress.  A 
stress  is  always  a  two-sided  phenomenon.  Every  force,  in 
fact,  is  one  of  a  pair  of  equal  and  opposite  ones  —  one 
component,  that  is,  of  a  stress,  either  like  the  stress 
exerted  by  a  stretched  elastic  cord,  which  pulls  the  two 
things  to  which  it  is  attached  with  equal  force  in  opposite 
directions,  and  which  is  called  a  tension  ;  or  like  the  stress 
of  a  pair  of  railway  buffers,  or  of  a  piece  of  compressed 
india-rubber,  which  exerts  an  equal  push  each  way,  and  is 
called  a  pressure.1 

When  an  elastic  cord  supports  a  weight,  the  stress  on 
the  cord,  called  tension,  is  equal  in  both  directions.  A 
stone  attracts  the  earth  with  the  same  force  that  the  earth 
attracts  the  stone.  The  existence  of  a  stress  in  a  medium 
in  the  case  of  gravitation  has  not  yet  been  demonstrated, 
but  it  is  thought  to  exist.  The  action  between  a  magnet 
and  a  piece  of  iron  is  a  mutual  action,  and  the  medium  for 

1  Lodge's  Mechanics,  p.  55. 


50  MECHANICS. 

the  exertion  of  this  magnetic  attraction  is  found  to  be  in  a 
state  of  strain.  The  magnet  cannot  exert  a  pull  on  the 
iron  any  greater  than  the  iron  exerts  on  the  magnet.  Two 
men  pull  at  the  opposite  ends  of  a  rope.  The  stress  in 
the  rope  is  obviously  in  both  directions.  It  is  no  less  cer- 
tainly so  when  one  end  of  the  rope  is  tied  to  a  post  while 
one  man  pulls  at  the  other  end. 

The  same  principle  applies  in  cases  where  motion  ensues. 
The  centripetal  force,  which  causes  a  rotating  body  to 
describe  a  circle  with  uniform  velocity,  is  applied  to  the 
rotating  body  through  the  intermediary  of  the  connection, 
visible  or  invisible,  of  the  body  with  the  centre.  Corre- 
sponding to  this  is  the  opposite  phase  of  this  stress  or  the 
reaction,  called  the  centrifugal  force.  This  is  not  a  force 
acting  on  the  rotating  body,  but  the  reaction  which  the 
rotating  body  exerts  on  the  centre.  The  centripetal  and 
centrifugal  forces  are  the  opposite  phases  of  the  stress 
between  the  rotating  body  and  the  centre  of  rotation. 

When  any  mutual  action  takes  place  between  two  bodies 
the  momenta  generated  in  opposite  directions  are  equal; 
but  the  velocities  are  not  equal  unless  the  masses  are  equal. 

With  unequal  masses  the  velocities  are.  inversely  as  the 
masses.  While,  therefore,  the  momentum  of  the  gun  is 
equal  to  that  of  the  bullet,  its  velocity  is  very  much  less. 

Considered  only  with  respect  to  one  portion  of  a  system 
of  bodies  a  stress  is  called  action ;  with  respect  to  the 
remainder  d£  the  system  it  is  called  reaction.  The  third 
law  states  that  these  two  phases  of  a  stress  are  always 
equal  and  in  opposite  directions. 

PROBLEMS. 

1.  A  gun  weighing  3,000  kilos,  and  placed  upon  a  smooth  plane 
discharges  a  SO  kilogramme  ball  at  an  elevation  of  30°.  Find  the 
velocity  of  the  gun's  recoil. 


KfXETICS.  51 

2.  An  inelastic  mass  of  900  kilos.,  moving  with  a  velocity  of  30 
metres  a  second,  meets  an  equal  and  similar  mass  moving  10  metres 
a  second  in  the  opposite  direction.  Find  the  velocity  of  the  total 
mass  after  impact. 

43.  Work  defined  (B.,  88;  M.  and  M.,  101;  D.,  39; 
Stewart's  Conservation  of  Energy).  —  "  Work  is  the  act 
of  producing  a  change  of  configuration  in  a  system  in 
opposition  to  a  force  which  resists  that  change."  Thus 
when  a  weight  is  lifted  from  the  earth  a  change  in  the  con- 
figuration of  the  weight  and  the  earth  is  produced  in  oppo- 
sition to  the  force  of  gravity  which  resists  the  change. 

Work  is  measured  by  the  product  of  the  force  and  the 
displacement  produced  in  the  direction  of  the  force.  The 
amount  of  work  is  expressed  as  the  product  of  two  num- 
bers, which  represent  respectively  a  force  and  a  space. 

Thus  W=  Fs  ; 

and  since  force  equals  the  product  of  mass  and  acceleration, 

W=  mas. 

When  the  displacement  produced  is  not  in  the  line  of 
action  of  the  force,  but  makes  an  angle  a  with  that  direc- 
tion, then  -* 
W=  Fs  cos  a. 

This  may  be  described  as  the  product  of  the  force  and 
the  component  of  the  displacement  in  the  direction  in 
which  the  force  acts ;  or  the  product  of  the  displacement 
and  the  component  of  the  force  in  the  direction  of  the  dis- 
placement. In  one  case  it  is  the  product  of  the  force  and 
the  effective  displacement;  in  the  other  the  product  of 
the  displacement  and  the  effective  component  of  the  force. 

There  is  no  work  done  by  any  force  unless  there  is 
actual  motion  produced.  Gravity  does  no  work  upon  a 
weight  at  rest;  it  does  work  upon  a  falling  weight. 


52  MECHANICS. 

The  unit  for  the  measurement  of  work  in  the  C.G.S. 
system  is  the  erg.  It  is  the  work  done  by  a  dyne  through 
a  distance  of  one  centimetre. 

Gravity  gives  to  a  gramme  a  velocity  of  approximately 
980  cms.  per  second.  It  is  therefore  equal  to  980  dynes. 
Hence  if  one  gramme  be  lifted  vertically  one  cm.,  the 
work  done  against  gravity  is  980  ergs ;  or,  one  erg  of  work 
is  done  in  lifting  g-J^-  gramme  one  cm. 

Since  work  is  the  product  of  force  and  distance,  it  fol- 
lows that 

F       W 

F-    7' 

or  force  is  the  linear  rate  of  doing  work.     The  multiples  of 
the  erg  sometimes  employed  are 

The  megalerg,  or  106  ergs. 

The  joule,  or         107  ergs. 

Power  or  activity  is  the  time-rate  of  doing  work. 

The  unit  of  power  employed  in  the  C.G.S.  system  is  the 
watt,  which  equals  107  ergs  per  second. 

A  horse  power  is  a  unit  in  the  gravitational  system,  and 
is  a  rate  of  doing  work  equal  to  33,000  foot-pounds  per 
minute,  or  550  foot-pounds  per  second. 

To  convert  this  into  watts  it  is  necessary  to  multiply  the 
550  by  the  ratio  between  a  foot  and  a  cm.,  then  by  the 
ratio  between  a  pound  and  a  gramme,  which  gives  gramme- 
centimetres;  and,  finally,  by  the  value  in  ergs  of  one 
gramme-centimetre,  or  980.  Hence 

550  x  30.4797  x  453.59  x  980  =  746  x  107  ergs  per  sec- 
ond, or  746  watts. 

One  horse  power  is  therefore  equivalent  to  746  watts. 

44.     Graphical  Representation  of  Work  (B.,  89).- 
Since  work  is  the  product  of  force  and  space,  it  is  evident 


s  cos  a 


-X 


KINETICS.  53 

that  work  may  be  represented  numerically  by  an  area. 
Thus  when  the  force  is  of  constant  value  the  work  done 
may  be  represented  by  the  area  of  a  rectangle,  one  side  of 
which  is  numerically  equal  to  the  force  and  the  adjacent 
side  to  the  distance,  or  the  component 
of  the  distance  in  the  direction  of  the 
force  (Fig.  23). 

If  the  force  increases  from  zero  to 
a  final  value  F,  then  the  work  done 
during  this  increase  is  the  product  of 
the  mean  value  of  the  force  and  the  displacement.    It  may 
then  be  represented  by  the  area  of  a  right  triangle  (Fig.  24), 
in  which  the  base  is  the  effective 

-i  r 

t  displacement,  and  the  altitude  the 
final  value  of  the  force  F ;  for  work 
then  equals  %Fs  cos  a,  which  is  the 

area  of  the  triangle. 

s  cos  a  __  .  •       ,  i 

— X       But    in    many   cases,    as    in   the 

cylinder  of  a  steam-engine,  the  force 

or  pressure  varies  according  to  a  more  complex  lawr.  If  p 
is  the  pressure  per  unit  area  of  the  piston,  and  A  is  the 
area,  then  the  total  pressure  is  P  =  pA. 

Let  now  x  be  a  small  distance  through  which  the  piston 
moves  while  the  pressure  p  remains  constant,  then  the 
work  done  on  the  gas  during  compression  is 

w  =  pAx. 

But  Ax  is  the  diminution  in  the  volume  of  the  gas 
which  we  may  denote  by  v. 

Then  w  =  pv, 

or  an  element  of  the  work  done  during  a  small  movement 
of  the  piston  is  the  product  of  the  pressure  per  unit  area 
and  the  small  change  in  volume. 


54  MECHANICS. 

The  total  work  done  during  a  finite  compression  of  the 
gas   may  be  represented  by  the  area   of  the   figure   ABba 
(Fig.  25),  in  which  the  ordinates  of  the  curve  AB  are  the 
successive   values   of   the   pressure, 
and  the  abscissas  represent  the  cor- 
responding  volumes  of  the  gas.  Take 
any  small  element  of  this  area  at 
Aa.     Then  the  length  of  this  strip 
is  the  instantaneous  pressure  jt?,  and 


b  n  its    width  is  the   indefinitely  small 

change  of  volume  v.    Hence  its  area 

is  the  element  of  the  work  done  w,  and  the  sum  of  all 
such  elements  is  the  total  work  done  on  the  gas  during  the 
compression.  This  is  the  area  ABba. 

45.  Energy  (L.,  79 ;  T.  and  T.,  73).  —  Whenever  work 
is  done  on  a  body  or  a  system  of  bodies,  so  as  not  merely 
to  heat  it,  but  in  such  a  manner  as  to  change  the  relative 
positions  of  its  parts,  then  there  has  been  conferred  upon 
the  system  the  capacity  of  doing  work  in  its  turn.  If,  for 
example,  a  mass  of  gas  is  compressed  by  a  piston  in  a 
cylinder,  work  is  done  upon  it  as  already  explained.  But 
the  gas  has  now  in  turn  acquired  the  ability  to  do  work  on 
the  piston,  both  because  of  its  higher  temperature  and  the 
increased  pressure  to  which  it  is  subjected. 

So  when  a  steam-engine  has  lifted  the  weight  of  a  pile- 
driver,  it  has  done  work  on  it  against  gravity.  In  its  new 
position  relative  to  the  earth  this  weight  has  the  capability 
of  doing  work  ;  and  when  it  is  released,  it  descends,  over- 
comes the  resistance  offered  by  the  pile,  and  forces  it  into 
the  ground. 

When  water  is  pumped  up  into  an  elevated  reservoir 
work  is  done  upon  it ;  but  the  water  thereby  acquires  the 


KINETICS.  55 

power  of  doing  work,  or  of  overcoming  resistance,  by 
means  of  a  proper  motor  mechanism.  It  possesses  some- 
thing which  it  did  not  have  at  the  lower  level. 

Work  may  be  done  upon  a  storage  battery  by  means  of 
a  steam-engine  and  a  dynamo-machine.  The  charged  bat- 
tery then  has  conferred  upon  it  the  power  of  doing  work 
by  the  capacity  which  it  has  of  furnishing  a  current  of 
electricity  to  run  an  electric  motor.  It  may  churn  the  air 
by  a  fan,  operate  a  printing  press,  run  a  street  car,  or  pro- 
pel an  electric  launch. 

Consider  some  examples  of  a  somewhat  different  char- 
acter. Work  is  done  upon  a  cannon  ball  by  means  of 
the  pressure  of  the  gases  arising  from  the  explosion  of  the 
powder.  The  ball  acquires  a  high  speed.  It  acquires 
more  than  that.  It  now  possesses  the  capacity  of  over- 
coming resistance.  By  virtue  of  its  mass  and  its  motion 
it  may  demolish  fortifications,  pierce  armor,  or  imbed  itself 
in  the  nickel-iron  plates  of  an  ironclad. 

When  work  is  done  by  the  steam  on  the  piston  of  an 
engine,  the  heavy  flywheel  is  made  to  revolve  on  its  axis. 
Work  is  done  upon  it  to  give  it  motion.  If  the  steam  is 
shut  off  the  engine  will  continue  to  revolve  and  may  do 
work  to  the  extent  to  which  the  massive  flywheel  now 
possesses  the  power  of  doing  it. 

In  all  these  cases  while  work  is  being  done  upon  the 
body  energy  is  transferred  from  the  active  agent  or  the 
source  to  the  system  upon  which  work  is  done.  The  sys- 
tem or  body  which  has  acquired  the  power  of  doing  work  is 
said  to  possess  energy.  Strictly  only  the  available  energy 
is  the  power  of  doing  work  or  of  producing  physical  changes 
in  other  bodies ;  for  a  body  may  possess  energy,  in  the 
form  of  heat  for  example,  which  cannot  be  made  to  do 
work,  or  is  not  available.  Work  may  now  be  defined  as 


56  MECHANICS. 

the  act  of  transferring  energy  from  one  body  or  system  to 
another. 

46.  Potential  Energy.  —  In  the  first  three  illustrations 
of  the  last  section  the  body  acquiring  the  energy  exerts 
force  or  pressure.     The  compressed  gas  presses  against  the 
piston ;  the  force  of  gravity  pulls  the  weight  of  the  pile- 
driver  downward  against  the  detent  holding  it ;  and  the 
electric  pressure    of   the   battery  is   ready   to   produce  a 
current  as  soon  as  the  circuit  is  closed.     During  the  trans- 
fer of  energy  from  the  working  agent  work  is  done  against 
this  force  or  pressure.     The  energy  thus  acquired  is  called 
potential  energy.     Potential  energy  is  energy  of  stress.     It 
is  often  called  energy  of  position  or  static  energy.     In  all 
cases  of  potential  energy  stress  or  force  is  present,  but  not 
motion ;  or  at  least  the  motion  forms  no  part  of  the  poten- 
tial energy.     The  energy  of  an  elevated  mass,  of  bending, 
twisting,  deformation,  of  chemical  separation,  arid  of  the 
ether-stress  in  a  magnetic  field  are  all  cas#s  of  potential 
energy. 

47.  Kinetic  Energy.  —  In  the  last  two  illustrations  of 
Art.  45  the  effect  of  doing  work  upon  the  body  is  to  give 
it  motion.     The  energy  which  it  thus  acquires  is  called 
kinetic  energy  or  energy  of  motion.     Motion  is  the  essen- 
tial condition  in  a  body  possessing  kinetic  energy.     Stress 
has  no  place  in  kinetic  energy.     But  if  a  resistance  is  sup- 
plied work  is  done  against  it  and  the  body  loses  kinetic 
energy. 

When  work  is  done  both  kinds  of  energy  are  usually 
present,  and  one  of  these  forms  of  energy  is  passing  into 
the  other  while  the  work  continues.  Thus  the  energy  of 
steam  in  a  boiler  is  partly  potential,  depending  upon  pres- 


KINETICS.  57 

sure,  and  partly  kinetic,  as  heat;  but  the  work  that  it 
does  in  the  cylinder  of  an  engine  consists  entirely  in  the 
immediate  conversion  of  potential  energy  into  kinetic 
energy,  or  energy  of  stress  into  energy  of  motion.  Where- 
ever  the  motion  is  with  the  force  speed  increases,  or  poten- 
tial energy  becomes  kinetic ;  when  the  motion  is  against 
the  force  kinetic  energy  becomes  potential.  Thus  if  a 
bullet  be  fired  vertically  upward,  the  motion  is  against  the 
force  of  gravity,  and  the  potential  energy  increases  at 
the  expense  of  the  kinetic ;  when  the  bullet  reaches  its 
greatest  elevation  its  energy  is  all  potential.  After  that 
it  descends,  the  motion  is  with  the  force,  and  the  potential 
energy  is  again  converted  into  the  kinetic  form. 

The  earth  travels  around  the  sun  in  an  elliptical  orbit 
with  the  sun  at  one  focus.  When  the  earth  is  nearest  the 
sun,  or  at  perihelion,  it  has  its  greatest  orbital  velocity 
and  greatest  energy  of  motion.  As  it  recedes  from  the 
sun  and  passes  toward  aphelion,  its  acceleration  toward 
the  sun  is  not  perpendicular  to  its  path.  This  accelera- 
tion may  be  resolved  into  a  normal  and  a  tangential  com- 
ponent*. The  first  produces  the  curvature  of  the  orbit, 
while  the  tangential  component  is  a  negative  acceleration' 
from  perihelion  to  aphelion.  The  speed  of  the  earth  there- 
fore diminishes,  and  its  kinetic  energy  decreases,  all  the 
way  from  perihelion  to  aphelion.  But  its  potential  energy 
at  the  same  time  increases  because  the  distance  from  the  sun 
increases.  At  aphelion  the  kinetic  energy  is  a  minimum 
and  the  potential  a  maximum.  The  opposite  transforma- 
tion then  takes  place  during  the  motion  from  aphelion  to 
perihelion. 

48.    Energy  is  not  Force.  —  It  is  a  very  common  error 
to  confuse  energy  and  force.     They  are  perfectly  distinct 


58  MECHANICS. 

and  represent  different  physical  concepts.  When  work  is 
done  on  a  body,  so  as  to  increase  its  available  energy,  the 
quantity  of  work  done  is  equal  to  the  gain  in  energy. 
Energy  is  therefore  measured  by  work,  and  the  unit  of 
measurement  is  the  erg,  the  same  as  the  unit  of  work. 
The  measure  of  energy  always  requires  two  factors,  one  of 
which  only  is  a  stress.  Energy  is  always  the  product  of 
one  of  several  pairs  of  factors.  These  pairs  do  not  always 
include  force.  Force  is  the  space-rate  of  transferring 
energy,  and  is  not  the  energy  itself.  Energy  cannot  be 
created,  increased,  or  diminished  by  any  natural  or  me- 
chanical process.  Force  may  be  augmented  to  any  extent 
by  numerous  mechanical  devices.  When  a  mass  of  matter 
moves  without  encountering  opposition  or  resistance  it 
carries  with  it  a  definite  quantity  of  kinetic  energy,  but 
exerts  no  force.  It  produces  neither  motion  nor  change 
of  motion  till  it  transfers  its  energy  to  another  body. 
Force  is  manifested  only  during  this  transfer. 

49.  Kinetic  Energy  in  terms  of  Mass  and  Velocity. 
—  In  uniformly  accelerated  motion 

v2  =  2as. 

Whence  %mv2  =  mas. 

But  mass  multiplied  by  acceleration  is  force.  The 
second  member  of  the  last  equation  is  therefore  the  prod- 
uct of  force  and  distance,  or  work.  It  represents  the 
work  done  upon  a  mass  m  to  give  to  it  the  velocity  v 
while  working  through  the  space  s;  and  as  the  energy 
stored  up  is  measured  by  the  work  done  in  its  production, 
it  follows  that  the  energy  of  the  mass  m,  moving  with  a 
velocity  v,  is  %mv*.  Or  conversely,  a  body  m,  moving  with 
a  velocity  v,  can  do  work  against  a  resistance  ma,  through 
a  space  s,  before  coming  to  rest. 


KINETICS.  59 

The  same  expression  for  the  kinetic  energy  may  be 
reached  in  another  way.  If  a  force  F  acts  on  a  mass  m 
during  an  interval  of  time  t,  then  the  measure  of  the  effect 
is  the  impulse,  or  Ft. 

By  the  second  law  of  motion  impulse  is  measured  by  the 
change  of  momentum.  If  the  initial  and  final  velocities 
of  m  are  VQ  and  v,  then  the  change  in  momentum  is 
m  (v  —  VQ) .  Therefore 

Ft=m(y-v^ (a) 

In  uniformly  accelerated  motion  the  mean  velocity  is 

£  (v  -f  v0).       It  is  also  —     Hence 

-t  -K»  +  O CO 

Multiply  00  and  (5)  together,  member  by  member,  and 
we  have 

Fs  =  \m  (v~  —  ?'02)  =  %mv2  —  %mv02. 

But  again  Fs  is  the  work  done  or  the  energy  accumulated 
in  the  moving  mass.  The  increase  in  the  kinetic  energy  is 
therefore  one-half  the  product  of  the  mass  and  the  differ- 
ence of  the  squares  of  the  initial  and  final  velocities.  If 
the  initial  velocity  is  zero,  then 

Fs  =  *mv- 
as  before. 

PROBLEMS. 

1.  A  truck  weighing  2,000  kilos,  moves  uniformly  along  a  level 
surface  with  a  velocity  of  20  metres  a  second.  Compute  its  kinetic 
energy. 

2.     Compare  the  kinetic  energy  of  a  ball  having  a  mass  of  15  gms 
and  a  velocity  of  400  metres  a  second  with  that  of  the  gun  from 
which  it  was  fired  if  the  mass  of  the  gun  is  8  kilos. 

5O.  The  Conservation  of  Energy  (M.  and  M.,  103; 
Stewart's  Conservation  of  Energy,  82).  —  In  the  ad- 
mirable language  of  Maxwell,  the  principle  of  the  conserva- 


60  MECHANICS. 

tion  of  energy  is  as  follows  :  "  The  total  energy  of  any 
material  system  is  a  quantity  which  can  neither  be  in- 
creased nor  diminished  by  any  action  between  the  parts 
of  the  system,  though  it  may  be  transformed  into  any  of 
the  forms  of  which  energy  is  susceptible."  If  some  agent 
external  to  the  system  does  work  upon  it,  then  its  energy 
is  increased  by  the  amount  of  work  done ;  if  the  system 
does  work  on  any  external  resisting  agent,  then  the  sys- 
tem loses  energy  equal  in  amount  to  the  work  done  by  it. 
But  if  we  include  the  given  system  and  the  external  agent 
in  one  larger  system,  the  energy  of  the  total  system  is 
neither  increased  nor  decreased  by  any  action  between 
them.  By  taking  into  account  different  parts  of  the 
physical  universe  successively,  we  finally  reach  the  conclu- 
sion that  the  quantity  of  energy  in  the  universe  cannot  be 
increased  or  diminished  by  any  action  of  which  we  have 
knowledge.  "  The  doctrine  of  the  Conservation  of  Energy 
is  the  one  generalized  statement  which  is  found  to  be  con- 
sistent with  fact,  not  in  one  physical  science  only,  but  in 
all.  When  once  apprehended  it  furnishes  to  the  physical 
inquirer  a  principle  on  which  he  may  hang  every  known 
law  relating  to  physical  actions,  and  by  which  he  may  be 
put  in  the  way  to  discover  the  relations  of  such  actions  in 
new  branches  of  science." 

The  great  doctrine  furnished  by  chemistry  is  the 
indestructibility  or  conservation  of  matter ;  the  great 
doctrine  furnished  by  physics  is  the  indestructibility  or 
conservation  of  energy.  These  two  entities,  matter  and 
energy,  comprise  all  of  the  physical  universe.  Matter  and 
energy  are  indestructible  by  any  power  save  that  of  the 
Almighty  who  created  them.  Is  it  preposterous  or  un- 
scientific to  put  intelligence,  the  only  other  entity  of 
which  we  have  knowledge,  in  the  same  category? 


KINETICS. 


51.  Transformations  of  Energy.  —  While  energy  is  in- 
destructible it  may  assume  almost  innumerable  forms,  either 
potential  or  kinetic.  A  few  examples  of  the  conversion 
from  energy  potential  to  energy  kinetic,  or  the  reverse, 
have  already  been  given.  But  all  physical  processes  involve 
energy  changes.  A  ceaseless  series  of  such  changes  is 
therefore  taking  place  in  the  orderly  course  of  nature. 
Moreover,  all  machines  are  only  instruments  for  the  trans- 
formation of  energy  and  the  turning  ^of  it  to  account  in 
effecting  useful  processes.  A  clock  or  a  watch  when  wound 
up  possesses  a  small  store  of  potential  energy  which  it 
gradually  expends  in  doing  the  work  of  turning  the  mechan- 
ism against  friction  and  the  resistance  of  the  air,  and  pro- 
ducing the  sound  of  ticking.  The  striking  mechanism  of 
a  tower  clock  receives  its  supply  of  energy  for  a  week's 
service  in  the  course  of  an  hour.  It  distributes  it  over  an 
entire  week  and  sends  it  out  hour  by  hour  over  an  area 
of  many  square  miles.  The  university  clock  requires 
134,000  foot-pounds  or  1816  x  109  ergs  of  energy  to  wind 
its  striking  side.  It  expends  it  by  484  strokes  daily, 
including  the  chimes,  or  a  total  of  3388  strokes  during 
the  week.  A  watch  in  the  same  time  divides  up  the  energy 
given  to  it  by  daily  windings  into  over  3,000,000  ticks. 

An  example  of  a  different  character  is  the  energy  in  the 
form  of  chemical  separation  which  is  put  into  a  rifle  car- 
tridge. The  pull  of  the  trigger  operates  to  convert  it  into 
the  kinetic  energy  of  the  bullet.  The  momentum  is  equally 
divided  between  the  bullet  and  the  gun,  but  not  so  with 
the  energy  ;  for  kinetic  energy  varies  as  the  square  of  the 
velocity.  The  kinetic  energy  therefore  goes  chiefly  with 
the  bullet. 

An  illustration  of  transcendent  interest  is  found  in  the 
radiant  energy  received  from  the  sun.  The  source  of 


62  MECHANICS. 

nearly  all  the  energy  which  the  earth  possesses  is  the  centre 
of  the  solar  system.  It  comes  to  us  through  the  medium 
of  the  ether  as  kinetic  energy.  Its  absorption  converts 
it  into  heat  and  warms  the  earth.  In  the  marvelous  chem- 
istry going  on  in  every  leaf  and  plant,  through  the  agency 
of  the  chlorophyll,  it  decomposes  the  carbon  dioxide  of 
the  air  and  water,  and  is  stored  up  in  the  cellulose  and 
woody  fibre  as  potential  energy.  In  this  form  it  may  serve 
as  food  for  man  and  animal,  or  may  remain  for  ages 
securely  locked  up  in  the  earth  as  coal  and  oil  to  furnish 
power  and  fuel  for  future  generations. 

In  combustion  or  decay  it  again  assumes  the  kinetic  form 
and  is  dissipated  beyond  recovery  as  heat. 

It  warms  the  tropical  ocean,  lifts  it  in  invisible  watery 
vapor,  transports  it  to  distant  continents,  and  pours  it 
down  as  rain  on  mountains  and  elevated  plains.  Thence 
it  descends  as  streams,  and  becomes  the  energy  of  water 
supplies. 

The  potential  energy  of  the  chemically  separated  fuel 
and  oxygen  may  become  mechanical  power  by  means  of 
the  boiler  and  the  engine.  The  potential  energy  of  the 
elevated  body  of  water  may  become  mechanical  energy  by 
descending  and  driving  the  turbine  wheel.  By  means  of  a 
dynamo-electric  machine  this  mechanical  energy  may  be- 
come the  kinetic  energy  of  an  electric  current.  The  con- 
ductor, or  the  ether  surrounding  the  conductor,  conveys  it 
to  distant  points  where  electric  lamps  again  convert  it  into 
light,  or  motors  utilize  it  for  any  of  the  purposes  to  which 
power  is  applied.  It  is  worthy  of  notice  that  electricity 
itself  is  not  energy,  but  is  only  the  medium  by  which  the 
energy,  which  in  every  case  has  been  derived  from  the 
sun,  is  distributed  arid  utilized  for  man's  comfort  and 
convenience. 


KINETICS.  63 

52.  The  Availability  of  Energy.  —  If  the  transforma- 
tions of  energy  are  attentively  considered  it  will  appear 
that  the  final  form  which  it  invariably  assumes  is  diffused 
heat.  Whenever  the  attempt  is  made  to  convert  kinetic 
into  potential  energy,  as  by  pumping  water  into  an  elevated 
reservoir,  or  by  storing  it  up  as  energy  of  chemical  separa- 
tion in  a  storage  battery,  the  conversion  is  always  incom- 
plete and  partial. 

It  is  impossible  by  any  means  at  our  command  to  effect 
a  complete  conversion  of  kinetic  energy  into  the  potential 
form.  Some  of  it  always  escapes  as  heat,  either  through 
friction,  radiation,  conduction,  the  heating  of  electric  con- 
ductors, or  by  some  of  the  means  by  which  energy  escapes 
during  conversion  and  transmission.  But  energy  in  the 
form  of  diffused  heat  is  not  available  for  further  use.  A 
heat-engine  requires  a  hot  body  and  a  cold  body  in  order 
that  heat-energy  may  be  utilized  while  passing  from  the 
hotter  body  to  the  cooler  one. 

In  the  same  way  all  the  processes  of  nature  exhibit 
energy-changes  o~"n  the  way  from  the  more  available  to  the 
less  available  state. 

Potential  energy  is  the  highly  available  form.  It  always 
tends  to  revert  to  the  kinetic  type,  but  in  such  a  way  that 
only  a  portion  of  the  kinetic  energy  is  available  to  effect 
useful  changes  either  in  nature  or  art.  Hence,  the  energy 
of  the  solar  system  is  becoming  all  the  time  less  and  less 
available. 

No  grander  survey  of  the  material  universe,  especially 
when  considered  from  the  point  of  view  of  energy-changes, 
has  ever  been  made  than  the  one  described  as  the  nebular 
hypothesis.  Each  celestial  system  is  considered  by  itself. 
The  nebular  hypothesis  traces  it  back  to  a  mass  of  widely 
disseminated  gaseous  matter.  It  may  not  even  at  that 


64  MECHANICS. 

period  be  faintly  luminous ;  it  possesses  only  the  inherent 
quality  of  gravitation,  and  therefore  of  potential  energy. 
Slowly  the  widely  extended  mass  gravitates  toward  its 
centre  of  mass.  The  parts  of  the  system  fall  toward  one 
another,  and  its  potential  energy  thereby  begins  to  suffer 
conversion  into  the  kinetic  energy  of  heat  and  light.  As 
long  as  the  falling  together  and  the  contraction  of  the 
mass  continue,  so  long  is  energy  of  position  transformed 
into  energy  of  motion.  It  is  easily  demonstrable  that  the 
temperature  of  the  contracting  mass  continues  to  rise  as 
long  as  it  remains  gaseous.  The  radiation  of  energy  into 
space  then  goes  on.  The  contraction  of  the  mass  means, 
therefore,  the  dispersion  of  the  energy.  The  potential 
energy  of  the  diffused  nebulous  matter  is  convertible  and 
available ;  the  converted  kinetic  energy  radiated  into 
boundless  space  is  chiefly  unavailable. 

Of  this  energy  radiated  from  the  sun  the  earth  re- 
ceives about  one  two-thousand-two-hundred-millionth  part 
(2.200,000,000)-  And  of  this  small  fraction  the  portion  stored 
up  in  vegetation  and  coal  is  only  an  infinitesimal  part. 
Nature  is  therefore  prodigal  of  energy,  not  economical ; 
and  the  small  store  that  the  earth  does  retain  ultimately 
becomes  diffused  and  radiated  heat. 

It  is  thus  apparent  that  the  energy  of  the  solar  system 
is  running  to  waste  or  becoming  unavailable. 

Unavailable  energy  is  called  Entropy.  The  inevitable 
conclusion  is  that  entropy  tends  toward  a  maximum. 

All  the  operations  of  nature  transform  energy  from  the 
available  into  the  unavailable  form.  Following  backward 
therefore  the  history  of  natural  operations  brings  us  to  a 
far-distant  period  when  all  the  energy  of  the  solar  system, 
and  indeed  of  the  physical  universe,  was  in  the  available 
form.  This  must  have  been  before  any  natural  operation 


65 

took  place.     "  In  the  beginning,"  then,  points  to  the  period 
when  all  energy  was  available. 

\Vith  no  less  certainty  physical  science  points  to  a  time 
when  entropy  shall  be  a  maximum.  All  the  processes 
of  nature  must  then  cease.  Even  the  earth  itself,  as  lifeless 
as  the  moon,  can  no  longer  circle  round  the  glowing  sun, 
but  both  and  all  together,  in  one  dead  mass,  must  hang  in 
everlasting  silence  in  the  boundless  night  of  space.  The 
marvellous  mechanism  will  then  have  run  down,  and  no 
further  motion  or  life-process  will  be  possible  unless  some 
new  order  intervenes  of  which  we  have  no  knpwledge  or 
conception. 

PROBLEMS. 

1.  A  mass  of  20  kilos,  moving  with  a  velocity  of  16  metres  per 
second,  overtakes  a  second  mass  of  32  kilos,  moving  with  a  velocity 
of   12  metres  per  second ;   find  the  common  velocity  after  impact. 
Calculate  the  loss  of  kinetic  energy,  both  bodies  being  inelastic. 

2.  To  a  vertical  axle  revolving  100  times  per  minute  there  is  at- 
tached at  its  lower  end  a  mass  of  75  kilos,  by  means  of  a  cord  1.5 
metres  long.     What  angle  does  the  cord  make  with  the  vertical,  and 
what  is  the  tension  in  the  cord  ? 

3.  A  tooth  in  the  blade  of  a  reaper  describes  a  simple  harmonic 
motion  of  4  cms.  amplitude  in  a  period  of  one-seventh  of  a  second. 
What  is  its  maximum  velocity  and  its  maximum  acceleration  per 
second  ? 

4.  Find  the  velocity  with  which  a  body  should  be  projected  down 
an  inclined  plane,  so  that  the  time  of  descending  the  plane  shall  be 
the  same  as  the  time  of  falling  through  the  vertical  height  of  the 
plane. 


u'G 


MECHANICS. 


CHAPTER  IV. 


KINETICS    (Continued}. 

53.  The  Moment  of  a  Force.  —  "  The  moment  of  any 
physical  agency  is  the  numerical  measure  of  its  impor- 
tance." The  moment  of  a  force  with  respect  to  a  point  is 
the  measure  of  the  tendency  of  the  force  to  produce  rota- 
tion about  the  point.  .  It  is  measured  by  the  product  of  the 

line  representing  the  force  and 
the  length  of  the  perpendicular 
drawn  from  the  point  to  the  di- 
rection of  the  force.  Let  0 
(Fig.  26)  be  the  point  about 
which  the  body  acted  upon  by 
the  force  AB  is  constrained  to 


Fig.  26. 


rotate.     The  moment  is  then 

Fx  OC=ABx  00. 

This  is  double  the  area  of  the  triangle  ABO.  The  line 
00  is  called  the  acting  distance  or  lever  arm,  and  the  point 
0  is  the  origin  of  moments.  The  moment  of  a  force  pro- 
ducing rotation  in  one  direction  is  positive,  and  the  mo- 
ment producing  rotation  in  the  other  direction  is  negative. 
In  the  figure  the  rotation  is  supposed  to  be  counter- 
clockwise. This  is  often  called  the  positive  direction.  A 
rotation  clockwise  would  therefore  be  negative.  It  is  not 
necessary  always  to  observe  this  conventional  rule. 


KINETICS. 


67 


E 


PROBLEMS. 

1.  ABCD  is  a  square  of  o  metres  on  each  side.    Forces  of  50,  60, 
and   70   dynes  act  along  AB,  AC,  and   AD.     Find   the  moment  of 
each  force  about  D. 

2.  Find  the  moment  of  each  of  the  forces  in  problem  1  about  the 
middle  point  of  AC. 

54.  The  Moment  of  the  Resultant  equals  the  Alge- 
braic Sum  of  the  Moments  of  the  Components.  —  Let 
the  origin  0  fall  outside 
the  angle  included  between 
the  lines  representing  the 
two  forces  P  and  Q.  The 
moments  are  all  then  of 
the  same  sign.  Complete 
the  figure  as  shown  in  the 
diagram  (Fig.  27).  Then  Fig  27 

p,  q,  and  r  are  the  acting 
distances  of  the  two  forces  and  their  resultant. 

From  the  similar  triangles  OAB'  and  ABE, 


AE_AB 

p    ~~  AO 
In  the  same  way 

AF  AC 
~q~  ~  AO 
AD 


or 


AB 


p 


or  AF  = 


Also 

But  AE  = 


~  AO" 

since 


or 


AO. 


AC  x  q 
~AO~ 
AD  x  r 


AO 

are  projections  on  the  same 


line  of  equal  and  parallel  lines.     Therefore 


Substitute  the  values  of  A  Gr,  AF,  and  AE,  and  multiply 
through  by  A  0.     Then 

ADxr  =  ABxp  +  ACx  q, 
or  R-r  =  P-p+  Q-q. 


68  MECHANICS. 

The  moment  of  the  resultant  equals  the  sum  of  the 
moments  of  the  components. 

If  0  lay  within  the  angle  BAC,  one  of  the  moments 
would  be  of  a  different  sign  from  the  others.  In  that  case 
the  moment  of  R  would  equal  the  arithmetical  difference 
of  the  other  two  moments. 

Since  the  proposition  is  true  for  two  forces  and  their 
resultant,  it  is  also  true  for  this  resultant  and  a  third  force, 
and  so  on  indefinitely.  It  is,  therefore,  true  for  any 
number  of  concurring  forces. 

A  case  of  particular  importance  occurs  when  the  origin 
0  is  on  the  line  of  the  resultant  or  this  line  produced. 

The  moment  of  the  resultant  is  then  zero  ;  and  the  sum 
of  the  positive  moments  is  equal  to  the  sum  of  the  negative 
moments,  or  the  tendency  to  rotate  in  one  direction  equals 
the  tendency  to  rotate  in  the  other  direction.  The  equa- 
tion of  equilibrium  is  then  written  by  placing  the  algebraic 
sum  of  the  moments  equal  to  zero. 

If  any  line  in  a  body  is  fixed  the  resultant  of  all  the 
forces  passes  through  this  line  if  there  is  no  rotation. 
Hence  the  sum  of  all  the  moments  with  respect  to  a  point 
on  this  fixed  line  is  zero. 

.iA/y 

55.  Parallel  Forces.  —  First,  two  forces.  Let  P  and 
Q  be  two  parallel  forces  in  the  same  direction  in  Fig.  28, 
and  in  opposite  directions  in  Fig.  29.  Then  by  Art.  20 
the  resultant  is  P  +  Q  in  the  first  case,  and  P —  Q  in  the 
second,  P  being  greater  than  Q.  Let  (7,  on  the  line  AB 
joining  the  points  of  application  of  the  two  parallel  forces, 
be  the  point  of  application  of  the  resultant.  Take  C  as 
the  origin  of  moments  and  draw  DE  perpendicular  to  the 
direction  of  P  and  Q. 


KINETICS. 


69 


Then  since  the  origin  is  on  the  line  of  the  resultant,  the 
moment  of  the  resultant  is  zero,  or 

Px  DC- 
P 

Q 


Whence 


DO 


But 


Therefore 


^ 

DO 
P_ 

Q 


^ 

AC 
RC 


or  the  lines  joining  the  points  of  application  of  the  two 

forces  with  that  of  the  resultant 
are  inversely  proportional  to  the 
forces. 
Also 

P  BC 

P+  Q  zs  BG+  AC* 


or  -  = 


and 


or 


Fig.  28. 
P 

P-Q 
P 
R 


BC 
AB 


(Fig.  28)  ; 


BC 


BO-AC 
BO 


AB 


(Fig.  29). 


B 


We  may  therefore  write 

P:  Q:  R::  BC : AC :  AB. 

We  may  then  say  that  of  two  par- 
allel forces  and  their  resultant,  each 
is  proportional  to   that   part  of   the 
line  joining  their  points  of  application  which  is   included 
between  the  other  two. 


Fig.  29. 


70  MECHANICS. 

Second,  any  number  of  parallel  forces.     Let  there  be  any 
number  of  parallel  forces,  P,  P',  P",  etc.     Then 

R  =  P  +  P'  +  P»  +  etc., 
each  force  with  its  proper  sign. 

Let  0  be  any  point  on  a  right  line  cutting  the  directions 
of  the  parallel  forces  at  right  angles,  and  let  #,  x',  x", 
etc.,  be  the  distances  from  the  intersections  of  the  forces 
with  this  line  to  the  point  0.  Also  let  X  be  the  distance 
of  0  from  the  intersection  of  the  resultant  R  with  the 
same  line.  Then  from  the  principle  of  moments 
EX  =  Px  +  P'x'  +  P"x"  +  etc. 

This  may  be  written 

RX=  ZPx, 

where  the  symbol  2'  means  "  the  sum  of  such  terms  as," 
all  the  terms  being  of  the  same  form. 

Finally 


This  determines  the  line  of  action  of  the  resultant. 


PROBLEMS. 

1.  A  stick  of  timber  of  uniform  cross-section  is  carried  by  three 
men,  one  at  one  end  and  two  by  means  of  a  bar  placed  crosswise 
under  the  stick.     Where  must  the  bar  be  placed  that  each  man  may 
carry  one-third  the  weight  ? 

2.  A  horizontal  bar  AB,  3  metres  long,  has  one  end  B  attached 
to  the  vertical  side  of  a  building.     The  other  end  A  is  supported  by 
a  rope  tied  to  it  and   to  the  building  at  a  point  4  metres  above  B. 
The  bar  supports  a  weight  of  50  kilos,  at  its  middle  point.     Find  the 
tension  in  the  rope  (by  moments). 

3.  Two  men  carry  a  weight  of  50  kilos,  slung  on  a  light  pole 
280  cm.  long.     If  the  weight  be  placed  at  a  distance  of  100  cm.  from 
one  end,  what  weight  does  each  man  carry  ? 


KINETICS.  71 

56.  Couples.  --  When  the  two  parallel  forces  are 
equal  and  oppositely  directed,  their  resultant  is  zero.  Such 
a  pair  of  forces  constitute  a  couple.  Their  resultant  is 
zero  so  far  as  motion  of  translation  is  concerned,  in  which 
all  points  of  the  body  move  in  parallel  straight  lines,  or 
any  line  drawn  within  the  body  remains  fixed  with  respect 
to  the  body  and  moves  parallel  to  itself. 


If  now  R  is  zero  the  first  member  of  the  equation  is  in- 
finity ;  therefore  BCis  infinite  in  value,  since  AB  is  not 
zero.  The  resultant  is  zero,  and  its  point  of  application  is 
at  an  infinite  distance. 

Such  a  pair  of  forces  may  cause  a  body  to  revolve  around 
an  axis.  The  moment  of  a  couple  is  the  product  of  one  of 
the  forces  arid  the  perpendicular  distance  between  their 
lines  of  action. 

One  couple  is  in  equilibrium  with  another  when  the 
moment  of  the  one  equals  the  moment  of  the  other,  and 
their  directions  of  rotation  are  opposite. 

PROBLEMS. 

1.  Four  forces  are  represented  in  direction,  magnitude,  and  line 
of  action  by  the  sides  of  a  square  taken  in  order  ;  prove  that  the  sum 
of  their  moments  about  every  point  of  the  square  is  a  constant. 

2.  Prove  that  the  forces  in  the  last  problem  are  equivalent  to  a 
couple. 

3.  Prove  that  the  moment  of  this  couple  is  numerically  equal  to 
twice  the  area  of  the  square. 

57.  The  Lever  --  A  lever  is  any  rigid  rod  or  bar,  the 
weight  of  which  may  be  neglected,  free  to  turn  about  a 
fixed  point  called  the  fulcrum.  The  problem  is  to  find  the 


72  MECHANICS. 

conditions  of  equilibrium  for  two  forces  tending  to  turn 
the  lever  in  opposite  directions  about  the  fulcrum. 

For  this  purpose  it  is  necessary  to  recall  that  a  result- 
ant is  a  single  force  which  will  produce  the  same  effects 
as  the  several  forces  compounded.  Hence  if  a  force  be 
applied  to  the  body  equal  to  the  resultant  of  all  the  forces 
acting  and  opposite  to  it  in  direction,  then  equilibrium  must 
ensue.  For  if  we  assume  the  several  forces  replaced  by 
their  resultant  and  a  force  applied  equal  and  opposite  to 
this  resultant  along  the  same  straight  line,  the  iinal  re- 
sultant would  be  zero,  or  no  motion  would  ensue.  Each 
force  would  then  be  equal  and  opposite  in  direction  to 
the  resultant  of  all  the  others. 

Let  two  weights,  P  and 
W  (Fig.  30),  be  applied  to 
the  two  ends  of  the  lever, 
and  let  the  lever  be  support- 
ed  at  the  point  of  application 
of  their  resultant  by  a  cord 
passing  over  a  pulley  with- 
out  friction,  and  carrying  a 
counter  weight  P  +  W 
Then  the  several  weights 

will  be  in  equilibrium.     Either  A,  B,  or   C  may  be  con- 
ceived to  be  fixed  and  the  equilibrium  will  be  maintained. 
Thus  if  C  be  fixed  it  becomes  the  fulcrum,  and  by  the 
principle  of  moments  (54) 

P__BC 
W  ~  AC\ 

the  moment  of  the  resultant  with  respect  to  C  being  zero. 
This  constitutes  a  lever  of  the  first  order. 

If  A  be  fixed  it  is  the  fulcrum,  the  resultant  of  W  and 
P  +  W  passes  through  A,  its  moment  is  zero, 


P~| 


and 


KINETICS.  73 

W         AC 


P+  W  "AS9 

or  power  and  weight  are  inversely  as  their  acting  distances. 

If  Wis  the  power,  the  lever  belongs  to  the  second  order. 

If  P-f  IF  is  the  power,  the  lever  is  of  the  third  order. 

The   relation  of   power  and   weight,  expressed    above, 
applies  to  the  several  orders  alike. 

58.    Principle   of  the  Lever  by  the  Theory  of  "Work. 
—  Suppose  the  system,  represented  by  Fig.  31,  to  suffer  a 
small  displacement,  so  that  A  and 
B  take  the  positions  A'  and  B'. 
Then  the  positive  work  done  on    A 
one  side  by  W  equals  the  nega- 
tive work  done  on  the  other  side 
against  P.  P\  r 

If    the  displacement  is  small,  Fig  3| 

the    arcs   A  A'  and   BB'  are  the 

distances  over  which  the  work  is  done  on  the  two  sides. 
Hence 

P-AA'=  W-BB', 


W~  AA' 

But  similar  arcs  are  to  each  other  as  their  respective 
radii.     Therefore 

P      BC 


or  power  and  weight  are  to  each  other  inversely  as  then 
lever  arms  or  acting  distances. 


74 


MECHANICS. 


32' 


59.  The  Inclined  Plane.  —  Since  accelerations  are 

proportional  to  forces,  the 
relation  already  found  to 
exist  between  the  accelera- 
tion of  gravity  in  free  fall 
and  the  acceleration  down 
an  inclined  plane  also  ap- 
plies to  power  and  weight, 
when  the  power  is  applied 
parallel  to  the  face  of  the 
plane  AC  (Fig.  32). 

If  g'  is  the  acceleration  parallel  to  the  face  of  the  plane, 

g'  =  g  sin  a, 
or  mg'  =  mg  sin  a. 

But  mg'  is  the  force  or  weight  P  and  mg  is  the  weight 
W.     Hence 

P  =  TFsin  a. 

Substituting  P  —  W—  ,  or  ----  =  - . 

The  same  result  may  be  obtained  by  means  of  the  prin- 
ciple of  work,  for  the  work  done  by  P  along  the  whole 
face  of  the  plane  A  C  equals  the 
work    done    against   gravity   in 
lifting   W  through  BC.     Hence 

P-l=W-  h, 


' 


or 


W 


B 


W 

Fig.  33. 


If  the  power  is  applied  par- 
allel to  the  base  of  the  plane 
(Fig.  33),  then  its  component  parallel  to  the  face  of  the 


KINETICS.  75 

plane  must  equal  the  component  of  W  in  the  same  direc- 
tion, or 

P  cos  a  =  IF  sin  a. 

P      sin  a  h 

Hence  -=- =  -     -  —  tan  a  —  -. 

W      cos  a  .    b 

If  the  principle  of  work  is  applied,  then  the  distance 
worked  over  must  be  multiplied  by  the  component  of  the 
force  in  the  direction  of  the  displacement  to  find  the  work 
done.  Again  assuming  that  the  weight  is  moved  from 
the  bottom  to  the  top  of  the  plane,  we  have 
P  cos  al  =  Wh. 

But  cos  a  —  _. 

L 

Hence  P-l=Wh, 

l 

P        h         ,    - 
or  —  =  -,  as  betore. 

The  inclined  plane  serves  to  increase  the  time  of  doing 
a  given  amount  of  work,  or  conversely  of  decreasing  the 
rate  of  doing  the  work,  that  is,  decreasing  the  power  re- 
quired. In  the  approaches  to  the  St.  Gotthard  tunnel  the 
necessary  rise  to  reach  a  given  elevation  is  distributed 
over  a  long  spiral  inclined  plane,  lying  partly  within  the 
mountain  as  a  tunnel  and  partly  on  its  face,  and  of  such 
length  that  the  power  required  to  lift  the  train  is  reduced 
to  that  which  the  engines  can  supply. 

60.    The  Sensibility  of  the  Balance  (A.  and  B.,  76).  - 
The  balance  is  an  instrument  for  the  comparison  of  equal 
masses  of  matter.     It  consists  of  a  light,  trussed  beam,  so 
as  to  have  the  required  stiffness  with  the  least  weight,  and 
it  is  supported  at  its  middle  point  by  means  of  a  "  knife- 


76 


MECHANICS. 


A 


Fig.  34. 


P+p 


edge  "  on  agate  planes.  At  each  end  is  suspended  a  scale 
pan,  and  the  two  pans  should  be  of  equal  weight. 

Let  the  three  points  A, 

A  B,  and  C  (Fig.  34)  be  in 

the  same  straight  line.  A 
and  B  are  the  knife-edges 
supporting  the  scale  pans, 
and  C  is  the  knife-edge 
on  which  rests  the  beam. 
The  centre  of  gravity  of 
the  beam  is  at  6r,  and  the 
weight  of  the  beam  is  w. 
Let  a  weight  P  be  placed 

in  one  scale  pan  and  P  +  p  in  the  other.  These  weights 
include  those  of  the  pans.  The  two  arms  of  the  balance 
should  be  equal  to  each  other.  Then,  upon  producing  a 
small  displacement,  so  that  the  beam  takes  the  position 
A'B',  the  two  lever  arms  remain  equal  to  each  other,  so 
that  P  and  P  have  equal  moments  and  counterbalance 
each  other.  They  may  therefore  be  omitted  from  the 
equation  of  equilibrium.  It  is  necessary  then  to  take 
into  account  only  the  moments  of  the  small  excess  weight 
p  and  the  weight  of  the  beam  u\  These  produce  rotation 
in  opposite  directions  around  the  fixed  point  C  as  the 
origin  of  moments. 

The  sensibility  of  the  balance  is  proportional  to  the 
angular  displacement  of  the  beam  with  a  given  small  dif- 
ference p  in  the  load. 

When  the  beam  assumes  its  position  of  equilibrium 
A'B',  the  moment  of  p  on  one  side  equals  the  moment  of  w 
on  the  other.  Therefore 

p  xB'I  =  wx  G'D. 


KINETICS.  77 

Let  CG-  be  called  r.  If  I  is  the  length  of  each  arm  of 
the  balance,  then  ^/equals  I  cos  0  and  Gr'D  equals  r  sin  6. 
Substituting 

p  X  I  cos  0  =  ^#  x  r  sin  0. 

From  which  K, 

xrx 


J3J.1J.      I/  ,  /J  //t/  •      . 

—a  =  tan  0  =  ¥—. 
cos  #  wr 


The  tangent  of  the  angular  displacement,  or  if  the  angle 
is  small  the  sensibility,  varies  directly  as  the  length  of  the 
beam,  and  inversely  as  the  weight  of  the  beam  and  the  dis- 
tance between  its  centre  of  gravity  and  the  knife-edge  or 
axis  of  suspension. 

If  the  three  points,  A,  B,  C,  are  not  in  the  same  straight 
line,  the  sensibility  changes  with  the  load.  When  C  is 
above  AB,  the  sensibility  is  diminished ;  for  when  the 
beam  is  displaced  the  lever  arm  of  the  higher  end  of  the 
balance  beam  becomes  longer  than  that  of  the  lower  end, 
and  the  moments  of  the  two  weights  P  and  P  become  un-  / 
equal  to  each  other.  The  difference  in  moments  increases 
with  P  and  tends  to  diminish  6. 

When  0  is  below  AB  the  sensibility  is  increased.     The    ' 
lever  arm  of  the  lower  end  of  the  beam  becomes  greater 
than  the  other,  and  the  difference  in  moments  increases  6. 

The  deflection  of  the  beam  under  load  raises  the  point 
C  with  respect  to  the  line  AB.  Hence  increase  of  load  may 
produce  first  an  increase  and  then  a  decrease  of  sensibility. 

61.  Double  Weighing.  —  The  process  of  double  weigh- 
ing serves  not  only  to  determine  the  true  weight  when 
the  arms  of  the  balance  are  unequal  in  length,  but  also  to 
determine  the  ratio  of  the  arms. 

Let  I  and  I'  be  the  lengths  of  the  arms  AC  and  BO. 


78  MECHANICS. 

Let  the  body  of  weight  w  be  first  suspended  from  A  and 
counterbalanced  by  a  weight  w'  in  the  other  scale  pan. 
Then  let  it  be  placed  in  B  and  counterbalanced  by 
weight  w". 

Then  for  the  first  operation 


For  the  second 


Multiplying  together  (a)  and  (5)  member  by  member 
ll'w^ll'w'w". 


Therefore  w  =  *w'  w". 

When  the  two  are  very  nearly  of  the  same  length,  and 
w'  differs  but  little  from  w",  the  true  weight  may  be  found 
very  approximately  by  taking  the  half  sum  of  w1  and  w". 

To  find  the  ratio  of  the  two  arms  it  is  convenient  to 
proceed  somewhat  differently.  Take  two  weights  of  the 
same  nominal  value  W  and  W  and  obtain  equilibrium 
with  one  in  each  pan.  Suppose  that  when  W  is  placed  in 
the  left-hand  pan  w'  must  be  added  to  it,  and  when  in  the 
right-hand  pan  w"  must  be  added  to  secure  a  balance. 
Then 

Z(TP+O  =  /'!*", 


Therefore  I2  W  (  W+  iv')  =  I12  W  (  W+  w"), 

or    ^  =  /— i  —  J*sr/  ^    J   =  1  +  U~~~ nearly.1 

+  «./    Vi  +  fV 

Both  w7  and  w"  must  be  taken  with  the  proper  sign. 


Kohlrausch's  Physical  Measurements. 


KINETICS.  79 


PROBLEM 

Left  Arm.  Right  Arm. 

W=  200  gms.  W  +  .008073 

PT'+.OOOo  PT=200 

Find  the  ratio  of  I  to  I'. 


62.  Centre  of  Inertia  (A.  and  B.,  44) If -we  con- 
sider any  system  of  equal  material  particles,  their  centre  of 
inertia  is  the  point  whose  distance  from  any  plane  is  the 
average  distance  of  the  several  particles  from  that  plane.. 

When  the  body  or  system  consists  of  a  finite  number  of 
parts  which  are  not  equal,  but  the  masses  of  which  are 
known,  then  the  distance  of  the  centre  of  inertia  of  the 
whole  body  from  any  plane  may  be  found  by  taking  the 
sum  of  the  products  of  the  several  masses  by  their  respec- 
tive distances  from  the  plane  and  dividing  by  the  sum  of 
the  masses.  The  centre  of  inertia  is  thus  the  average  dis- 
tance of  all  the  masses  from  the  plane. 

Conceive  now  the  system  of  particles  to  be  acted  on  by 
a  system  of  parallel  forces  proportional  to  the  masses  of 
the  particles.  Then  if  the  particles  be  supposed  collected 
at  the  centre  of  inertia,  or  centre  of  mass,  and  to  be  acted 
on  by  the  resultant  of  all  the  parallel  forces,  the  same  mo- 
tion will  be  produced  as  if  all  the  separate  forces  acted  on 
the  separate  particles.  The  motion  will  Be  motion  of 
translation  without  rotation. 

\Vhen  the  parallel  forces  are  due  to  gravity  the  point 
defined  above  is  called  the  centre  of  gravity.  The  centre 
of  inertia  and  the  centre  of  gravity  are  identical  when  the 
forces  of  gravity  are  strictly  parallel  and  proportional  to 
the  several  masses  constituting  the  body. 

The  "distance  X  of  the  centre  of  inertia  of  any  body  or 


80 


MECHANICS. 


system  of  bodies  from  a  given  plane  may  be  expressed  in 
the  form  of  a  general  formula, 

y  _  2mx 
2'm 

It  may  be  demonstrated  as  follows  :  Let  MM'  (Fig.  35) 
be  the  plane  of  reference.  Let 
the  particles  m^  m2,  be  situated  at 
A  and  B,  distant  respectively  xl 
and  x2  from  the  plane.  Let  C  be 
the  centre  of  mass  of  the  two  par- 
ticles ;  that  is,  the  centre  of  two 
parallel  forces  at  A  and  B  propor- 
tional to  mi  and  m2.  Through  C 
draw  DE  parallel  to  FQ-;  .Fand 
G-  are  the  points  of  intersection 
of  the  perpendiculars  from  m2  and 
mi  with  the  plane.  Let  x  be  the 
distance  of  C  from  the  plane. 


Lr{~' 

_    _ 

X 

/ 

71, 

C 

j} 

X2      W2J 

V 

Fig.  35. 


Then 


m_i_BO 
m2 


BE 


AC 

or  mi  x  AD  —  m2  x  BE. 

But  AD  =  Xi  —  #,  and  BE  =  x  —  x2. 

Hence        ,        mi  (xi  —  x)  =  m2  (x  —  xf). 
Transposing,     x  (ml  +  ra2)  =  miXi  +  m2  x2. 

The  same  process  may  be  applied  to  the  sum  of  mi  and 
m2  and  a  third  particle  m3,  and  so  on  indefinitely. 
may  therefore  write  generally 

Xlm  —  Zmx, 


We 


or 


X  = 


2'm 


KINETICS.  81 

If  in  the  figure  the  plane  of  reference  should  pass 
through  G7,  x  would  be  zero.  In  general  when  the  plane  of 
reference  passes  through  the  centre  of  inertia,  so  that  X 

is  zero,  we  have 

=  0. 


63.  Centre  of  Inertia  of  a  Triangle.  —  Let  the  triangle 
be  of  uniform  thickness  and  density.  Divide  it  into  a 
very  large  number  n  of  small  areas  by  equidistant  lines 
drawn  parallel  to  the  base.  From  the  apex  C  (Fig.  36) 
draw  CD  to  the  middle  point 
of  the  base.  It  passes  through 
the  middle  points  of  all  the  nar- 
row elements  of  the  triangle, 
which  are  their  centres  of  in- 
ertia. Their  centres  of  inertia 
all  lie  on  the  line  (7Z>,  and 
therefore  the  centre  of  inertia  A/  \  \B 

of  the  entire  triangle  must  lie  Fig.  36. 

on  the  same  line.    Let  it  be  at 

G-.     The  problem  is  to  find  the  distance  of   G-  from  the 
apex  C. 

The  areas  of  the  small  divisions  of  the  triangle,  begin- 
ning at  the  apex,  are  strictly  as  1,  3,  5,  etc.  But  no 
error  will  be  introduced  by  assuming  these  areas,  and 
therefore  their  masses,  to  be  represented  by  the  numbers 
1,  2,  3,  etc.  For  suppose  the  small  triangular  additions  to 
be  made  to  the  several  small  areas  as  shown  in  the  figure 
near  C.  The  areas  would  then  be  strictly  as  1,  2,  3,  etc. 
Now  since  there  is  an  infinite  number  of  divisions  n  of 
the  triangle,  the  base  and  altitude  of  the  equal  triangular 
additions  are  infinitesimal  quantities.  The  area  of  each 
triangle  is  therefore  an  infinitesimally  small  quantity  of 


82  MECHANICS. 

the  second  order.  The  error  made  then  by  adding  in  an 
infinite  number  of  these  excess  triangles  is  only  an  infini- 
tesimally  small  quantity  of  the  first  order,  and  this  is 
negligible  in  comparison  with  the  finite  area  of  the  large 
triangle. 

If  m  is  the  mass  of  the  first  small  area,  (lie  series  of 
masses  will  be  represented  by  the  numbers 
m,  2m,  3m,     .     .     .     nm. 

Let  the  plane  of  reference  be  drawn  through  C  perpen- 
dicular to  CD.  Then  the  products  mx  will  be  represented 
by  the  series 

I2m,  2X  3X     .     .     .     n2m. 

v      Zmx      m  (!2+22  +  32  +  .  w2) 

Hence  CG-  or  X—  -  } . 

2m       m  (1+2+3+    .    .    .  n) 

Apply  to  the  summing  of  the  two  series  the  formula  of 

m  +  1 

Art.  15,  s  =  -        —  ,  in  which  m  is  the  exponent. 
m  +  1 

2  +  1 
mi  -t»  rv>  o-2  2  n  ^ 

Then       1J  +  2-  +  32  +   .     .     .      n*  ==  —^  ^  ==  --  • 


n3          n2      2 
Hence  X  —  ra-—  ~  m—  =  -n. 

But  since  n  is  the  number  of   equal  divisions  of   CD, 

'>  2 

"n  equals   -CD. 
3  3 

PROBLEMS. 

1.  Show  that  the  centre  of  inertia  of  three  equal  weights  placed 
at  the  three  corners  of  a  triangle  corresponds  with  the  centre  of 
inertia  of  the  triangle  itself. 

2.  Find  the  centre  of  inertia  of  a  square  from  which  one  section 
made  by  the  two  diagonals  has  been  removed. 


KINETICS. 


83 


3.  A  uniform  circular  plate  has  a  circular  hole  cut  in  it  with  a 
diameter  equal  to  the  radius  of  the   plate,  the  two  circles  being 
tangent.     Find  the  centre  of  inertia  of  the  remainder. 

4.  A  square  is  described  on  the  base  of  an  isosceles  triangle. 
What  is  the  ratio  of  the  altitude  of  the  triangle  to  its  base  when  the 
centre  of  inertia  of  the  whole  figure  is  at  the  middle  point  of  the 
base? 

5.  If  two  triangles  have  the  same  base  and  equal  altitudes,  show 
that  the  distance   between  their  centres  of  inertia  is   one-third  the 
distance  between  their  vertices. 

64.  Centre  of  Inertia  of  a  Pyramid.  --  The  pyramid 
may  have  any  base.  Conceive  it  to  be  divided  into  small 
sections  by  a  very  large  number  n  of  equidistant  planes 
drawn  parallel  to  the  base.  Let  y  be  the  centre  of  inertia 
of  the  section  at  the  base,  and  draw 
Cg  (Fig.  37).  Since  the  sections  of 
the  pyramid  are  similar  figures,  the 
line  Cg  passes  through  the  centre 
of  inertia  of  all  these  elementary 
areas.  The  centre  of  inertia  of  the 
whole  pyramid  must  therefore  lie 
on  this  same  line.  Let  it  be  at  G-. 

To  find  CG-  suppose  a  plane  of 
reference  drawn  through  0  perpen- 
dicular to  Cg.  Then,  since  the  ele- 
mentary areas  of  equal  thickness  and  density  are  simi- 
lar figures,  these  areas  are  proportional  to  the  squares  of 
their  homologous  dimensions.  The  homologous  dimensions 
are  proportional  to  their  distances  from  (7,  that  is,  to 
1,  2,  3,  ...  n. 

If  now  the  mass  of  the  first  section  at  C  be  represented 
by  w,  then  the  masses  of  all  the  sections  may  be  repre- 
sented by  the  series 


Fig.  37. 


84  MEG  if  AN  res. 

The  products  of  these  masses  by  their  respective    dis- 
tances from  the  plane  of  reference  will  be  the  series 
I3m,  23m,  33w,     .     .     .     nsm. 

nri        ~      Imx       n4          n3         3 
Hence       CG-  =  X  =  -_  —  =  -m  —  -w  =  -n. 


2m        4       '    3          4 
The  centre  of  inertia  of  the  pyramid  is  therefore  on  the 
line  drawn  from  the  apex  to  the  centre  of  inertia  of  the 
base,  and  three-fourths  of  the  distance  from  the  apex  down. 

65.  Moment  of  Inertia  (A.  and  B.,  56  ;  L.,  46.) —  When 
a  body  rotates  every  point  of  it  describes  a  circle  around 
a  line  called  the  axis  of  rotation.  Every  point  of  the  body 
then  has  its  own  velocity  in  its  circle  of  rotation,  and  this 
velocity  is  proportional  to  the  distance  of  the  particle  from 
the  axis  of  rotation.  Hence  "  to  express  the  speed  with 
which  a  body  rotates,  it  is  sufficient  to  give  the  velocity 
of  any  one  point  together  with  its  distance  from  the  axis." 
The  distance  chosen  to  express  the  velocity  of  rotation,  or 
the  angular  velocity,  as  this  is  called,  is  unity.  Hence  the 
angular  velocit}*  of  a  body  is  the  linear 
velocity  of  a  point  situated  at  unit  dis- 
D  tance  from  the  axis.  This  is  represented 
by  the  Greek  letter  o>  (Fig.  38),  where 
rig.  as.  Oa  is  unity. 

Since  the  linear  velocity  of  any  par- 
ticle is  proportional    to   its    distance  from  the   axis,  the 

velocity  of  the  particle  m  at  A  is  v  =  rco,  or  o>  =  -,  where 

r  equals  OA. 

Again  if  the  velocity  is  uniform,  a  point  at  a  distance 
of  unity  from  the  axis  0  describes  a  circumference  2?r 
in  one  revolution  of  period  T.  Hence  angular  velocity  is 

w  =  ~~.     (Compare  Art,  33.) 


KINETICS.  85 

The  angular  velocity  represents  the  angle  turned 
through  per  second  by  the  whole  body,  as  well  as  the 
distance  travelled  by  a  particle  at  unit  distance  from 
the  axis  in  the  same  time.  It  must  not  be  overlooked  that 
the  angle  considered  is  always  measured  in  circular 
measure. 

Angular  velocity  may  be  variable.  Angular  acceleration 
is  then  the  time-rate  of  change  of  angular  velocity.  Let 
a  represent  angular  acceleration.  Then 


where  COQ  is  the  initial  and  co  the  final  angular  velocity,  the 
change  being  uniform  throughout  the  period  t. 
If  the  initial  velocity  is  zero,  then 


But  -  —  «,  the  linear  acceleration  of  a  point  at  a  dis- 

tance r  from  the  axis.     Hence 

a 
a  =  -  ,  or  a  =  ar  ; 

the   angular  acceleration  is  the    linear   acceleration  of   a 
particle  situated  at  unit  distance  from  the  axis. 

Let  the  particle  m  (Fig.  38)  be  at  A,  a  distance  r  from 
the  axis,  and  let  its  angular  velocity  be  oj.  Then  its 
linear  velocity  is  rco.  But  kinetic  energy  is  half  the  prod- 
uct of  the  mass  and  the  square  of  the  velocity.  The  kinetic 
energy  of  the  particle  is  therefore 


The  kinetic  energy  of   the  entire  rotating  body  is  the 
sum  of  all  such.  expressions  as  this  last  one,  or 


86  MECHANICS. 

Since  the  angular  velocity  is  the  same  for  all  points  of 
the  body,  or  is  a  constant  and  may  be  taken  out  from  the 
sign  of  summation.  The  quantity  Zmr  is  called  the 
moment  of  inertia  of  the  body.  It  measures  the  importance 
of  the  body's  inertia  with  respect  to  rotation.  It  is  pro- 
portional to  the  kinetic  energy  of  rotation.  The  work  done 
upon  a  body  to  give  it  an  angular  velocity  w  is  therefore  pro- 
portional to  the  moment  of  inertia  of  the  body.  The 
energy  of  rotation  of  a  body  whose  angular  velocity  is  a) 
depends  not  only  upon  its  mass,  but  upon  the  manner  in 
which  that  mass  is  disposed  about  the  axis. 

If  the  entire  mass  of  the  body  is  supposed  collected  at  a 
distance  Jc  from  the  axis  and  so  situated  that  the  moment 
of  inertia  remains  unchanged,  then 


The  distance  Jc  is  called  the  radius  of  gyration.  The 
moment  of  inertia  is  usually  represented  by  the  letter  /. 

66.  The  Moment  of  Inertia  and  Angular  Accelera- 
tion. —  The  moment  of  inertia  may  be  defined  as  the 
moment  of  the  couple  required  to  produce  unit  angular 
acceleration.  As  shown  in  the  last  section,  the  linear 
acceleration  a  of  a  particle  distant  r  from  the  axis  is  r  times 
the  angular  acceleration  or  ra.  Since  force  is  the  product 
of  mass  and  acceleration, 

f  —  mra. 

The  moment  of  this  force  about  the  axis  is  nw*a.  The 
acting  distance  is  r,  since  the  acceleration,  and  therefore 
the  force,  are  directed  tangentially  to  the  circle  in  which 
m  rotates.  The  total  moment  of  all  the  forces  producing 
the  rotation  of  the  entire  body  is  therefore 

Fb  =  a  (mr*  +  m'r*  +  m"r"*  +      ...)  =  almi*. 


KINETICS.  87 

If  now  a  is  unity,  then  the  moment  of  the  couple  producing 
unit  angular  acceleration  is  2W2,  or  the  moment  of  inertia. 
Also 

m 

a=  -v— j' 

2mr2 

in  which  b  is  the  lever  arm  at  which  the  force  F  acts  to 
produce  the  rotation. 

PROBLEM. 

The  weight  of  a  fly-wheel  is  M  gms.  Let  it  be  considered  entirely 
in  the  rim  at  a  distance  r  from  the  centre.  If  a  force  of  F  dynes 
acts  steadily  upon  the  wheel  at  an  arm  of  b  cms.,  show  that  the 
angular  velocity  w  after  t  seconds  from  the  commencement  of  the 
motion  will  be 

Fbt 
Mr* ' 

67.  To  find  the  Moment  of  Inertia  of  a  Rod.  —  Let 
the  axis  of  rotation  be  at  right  angles  to  the  length  of  the 
rod  through  its  middle  point.  Conceive  each  half  of 
the  rod  to  be  divided  into  n  equal  divisions,  each  of  mass 
m.  Then  the  moment  of  inertia  of  the  entire  rod  will  be 

7=-2m  (I2  +  22  +  32  +    .     .     .    O  =  2w-. 

o 

But  the  mass  of  the  rod  is  2mn.  Hence  if  Jtf"is  the 
entire  mass, 


7  jz 

The  length  I  of  the  rod  is  2n  or  n  —  -  ,  and  n2  =  -.    Sub- 

L.  4 

stituting  and 


Since  a  rectangle  may  be  conceived  to  be  made  up  of 
an  indefinite  number  of  such  parallel  rods,  if  an  axis  be 


88  'MECHANICS. 

supposed  to  bisect  two  opposite  sides  of  the  rectangle,  the 
moment  of  inertia  for  each  elementary  rod  will  be  the  ex- 
pression above.  If  then  m  is  the  mass  of  each  elementary 
rod  or  strip  and  a  the  length  of  the  rod,  or  the  bisected 
side  of  the  rectangle, 


If  b  is  the  other  side  of  the  rectangle,  the  moment  of 
inertia  for  an  axis  bisecting  the  sides  b  is 

*--& 

The  moment  of  inertia  of  a  rectangle  about  an  axis 
through  its  centre  of  figure  and  perpendicular  to  its  plane 
is  still  larger. 

68.    The  Moment  of  Inertia  of  a  Circle.  —  (a)  Let  the 
axis  be  through  the  centre  and  at  right  angles  to  the  plane 
of  the  circle.     Conceive  the  circle  to  be  made  up  of  ele- 
mentary rings  of  equal  width  and  of  radii 
1,  2,  3,      .      .      .     n, 
n  being  a  very  large  number. 

Let  the  mass  of  the  inner  ring  be  m  and  its  radius  unity. 
Then  the  masses  of  the  several  rings  will  be 

m,  2m,  3m,     .     .     .     nm, 

since  they  are  of  equal  width  and  their  circumferences 
are  proportional  to  their  radii.  The  density  is  of  course 
assumed  to  be  uniform.  Hence  we  shall  have  for  the 
moments  of  inertia  of  the  successive  rings 

Ira  •  I2,  2m  •  22,  3m  •  32,     .     .     .     nm-n2; 

and  1=  m  (I3  +  23  +  33  +     •     .     .     n3)  -  m^  . 

But        Jf=wl  +   2  +  3  +      .     .     .     n=mn* 


KINETICS.  89 


Therefore 

& 

Finally,  since  n  =  r,  the  radius  of  the  circle, 

1=  Mr* 
2 

'    Since  a  cylinder  is  made  up  of  such  rings,  the  moment 
of    inertia   of    the    cylinder   about   its    axis    also    equals 

M—  ,  in  which  M  is  the   mass  of  the  cylinder  and   r  its 
2i 

radius. 

(5)  To  find  the  moment  of  inertia  of  a  cylindrical 
shell  or  ring  about  its  axis,  let  r  and  r'  be  the  external  and 
internal  radii  ;  let  M  be  the  mass  of  the  shell  or  ring  and 
M  '  and  M  "  the  masses  of  the  cylinders  of  radii  r  and  r' 
respectively.  Then  the  moment  of  inertia  of  the  ring  is 


Now  M  '  —  7rr2ld  and  M"  =  7rr/2ld,  I  being  the  length  and 
d  the  density  of  the  cylinder.     Substitute  and 


But  Jf,  the  mass  of  the  cylindrical  shell,  is  irld  (r2  —  r/2). 

M 

Finally  then  1=  ~  (r2  +  r'2)  . 

(<?)  The  moment  of  inertia  of  a  right  cone,  around 
the  axis  of  figure,  may  be  obtained  in  a  similar  way. 

Divide  the  cone  into  thin  circular  laminae  by  equidistant 
planes  parallel  to  the  base.  The  radii  of  these  circles  are 
1,2,3,  .  .  .  n. 

The  masses  are  proportional  to  the  squares  of  the  radii. 
If  the  mass  of  the  first  one  is  m,  the  mass  of  the  whole 
cone  is 


MECHANICS. 


The  moments  of  inertia  of  the  circular  laminae  are 


m 


m     2  2 

— ?r  *  rr. 


Hence  J=  ^  (!4+24  +  34  +    .      .      .     rf)=5-i. 
2  J      5 

_..  „      i»» 

Substituting   Jf  for  -—  and 
o 

I=  10  n  ~~  10  r2' 

if  r  is  the  radius  of  the  base. 

PROBLEMS. 

1.  Find  the  moment  of  inertia  of  a  grindstone  1  metre  in  diameter 
and  10  cms.  thick,  density  2.14.     The  axis  is  through  its  centre  and 
perpendicular  to  its  plane. 

2.  Find  the   kinetic  energy  of  the  same  stone  when  rotating  5 
times  in  6  seconds. 

3.  A  homogeneous  cylinder  of  mass  m  and  radius  a  turns  round 
a  horizontal  axis  coinciding  with  the  axis  of  the  cylinder ;  a  fine 
thread  is  wrapped  round  it,  and  a  mass  m'  is  attached  to  the  end. 
Show  that  when  the  mass  m1  has  descended  through  the  height  h  the 
angular  velocity  of  the  cylinder,  neglecting  friction,  is  given  by  the 

equation  w2  = 


2m') 


G 


<z 

69.    Moment  of  Inertia  about  a  Parallel  Axis.  —  Let 
v  G-  be  the  centre  of  in- 

ertia or  centre  of  mass 
of  the  body ;  and  sup- 
pose the  moment  .of 
~~^  inertia  of  the  body 
about  the  axis  G-Yto 
be  known,  and  let  it 
be  denoted  by  /0.  To 
find  the  moment  of 
inertia  about  any  par- 
allel axis  through  A 
(Fig.  89),  the  distance  between  the  two  axes  being  d. 


Fig.   39. 


KINETICS.  91 

Let  an  element  of  the  mass  in  be  at  B.  GX^  G-Y,  GZ 
are  rectangular  axes.  Tlien  the  figure  GCBX  is  a  rec- 
tangle. In  the  triangle  ABG 

p*  =  r2  +  d2  -  2rd  cos  A  GB. 

Since  the  angle  at  0  is  a  right  angle, 
r  cos  A  GB  =  x. 

Therefore  p2  =  r2  +  d2  -  2dx, 

and  mp2  =  mr  +  md2  —  2dmx. 

For  every  element  of  the  mass  a  similar  equation  can  be 
written.     Therefore  summing  up  for  the  entire  body, 
Imp2  =  Imr2  +  2mcP  -  Ulmx. 

But  by  Art.  62  when  the  plane  of  reference,  which  is 
here  the  plane  ZGY,  passes  through  the  centre  of  inertia, 
2'mx  is  zero.  The  last  term  of  the  above  equation  is  then 
zero,  or  the  sum  of  all  the  positive  products  mx  is  equal  to 
the  sum  of  all  the  negative  ones.  Hence  if  /is  the  moment 
of  inertia  about  the  axis  through  A, 

I=I0  +  Md2, 
since  Im  equals  the  entire  mass  of  the  body. 

Therefore  the  moment  of  inertia  about  any  axis  exceeds 
the  moment  of  inertia  about  a  parallel  axis  through  the 
centre  of  inertia  by  a  quantity  equal  to  the  product  of 
the  mass  of  the  body  and  the  square  of  the  distance 
between  the  two  axes,  or  by  a  quantity  equal  to  the 
moment  of  inertia  which  the  body  would  have  about  the 
axis  through  A  if  its  mass  were  all  aggregated  at  its  centre 
of  gravity  G. 

PROBLEM. 

Two  cylinders,  of  radius  1.007  cms.  and  mass  119.6  gms.  each, 
are  placed  vertically  on  opposite  ends  of  a  bar  suspended  from  its 
middle  point  so  as  to  turn  freely  in  a  horizontal  plane,  the  distance 
from  the  centre  of  the  bar  to  the  centre  of  each  cylinder  being  11.45 


MECHANICS. 


cms.     Find  the  increase  in  the  moment  of  inertia  of  the  bar  due  to 
the  addition  of  the  cylinders. 

70.    The  Ideal  Simple  Pendulum By  an  ideal  simple 

pendulum  is  meant  one  in  which  the  entire  mass  is  sup- 
posed collected  at  a  point 
and  suspended  by  an  in- 
extensible  thread  without 
weight.  Let  the  mass  m  be 
suspended  from  A  (Fig. 
40)  by  the  thread  of  length 
I.  In  the  position  B  the 
acceleration  BG-  may  be 
*  resolved  into  two  compo- 
nents, the  one  in  the  direc- 
tion of  the  length  of  the 


string,  the  ineffective  com- 
ponent, and  the  other  tan- 
gential, the  effective  com.* 
ponent.     The  latter  is 
=     sm  0. 


Fig.  40. 


If   the   angle  BAN =  6 
is  small,  sin  6  may  be  put 

BN          x 

equal  to  6.     Moreover,  0  equals      -    or    — ,  the  displace- 
ment BN  being  called  x. 

Hence  ~f~(J^  —  (j^- 

'  I 

The  acceleration  of  the  mass  m  at  B  is  therefore  pro- 
portional to  its  displacement  x  from  the  middle  point  of 
its  path  N.  But  this  relation  is  characteristic  of  simple, 
harmonic  motion.  The  motion  of  the  pendulum  is  there- 
fore simple  harmonic  within  the  limits  of  the  approximation 
that  sin  0  =  6. 


KINETICS.  93 

In  simple  harmonic  motion 


Then  -  =  V>,    /=  iv,  and  T  =  -      . 


This  is  the    general  formula  for  the  period  of  oscilla- 
tion in  simple  harmonic  motion.     But  for  the  pendulum 


For  the  simple  pendulum  therefore  //.  =£ . 

Hence  T '=  2?r  ^  I  —  . 

This  is  the  period  of  a  complete  or  double  swing.     For 
a  single  vibration 


'The  periods  of  pendulums  of  different  lengths  are  pro- 
portional to  the  square  roots  of  the  lengths. 
If  T  is  one  second 


—  ? 
9 


or  g  =  7r2L 

But  TT  =  3.14159  and  vr2  =  9.869. 


Therefore  2=  4  =          ,  =  99.3  cms. 

TT"       9.869 

If  the  acceleration  of  gravity  were  986.9  the  seconds 
pendulum  would  be  exactly  one  metre  long. 

The  formula  for  the  period  of  vibration  is  independent 
of  the  displacement  x.  Within  the  limits,  therefore,  of  the 


94 


MECHANICS. 


approximation  that  the  sine  of  an  angle  is  equal  to  the 
angle  itself,  the  vibrations  are  isochronous,  or  the  period 
remains  the  same  while  the  amplitude  diminishes. 

PROBLEMS. 

1.  Find  the  period  of  oscillation  of  a  pendulum  6  metres  long 
at  a  place  where  g  is  980. 

2.  If  the  length  of  the  seconds  pendulum  is  99.414  cms.,  what  is 
the  value  of  g  ? 

3.  A  seconds  pendulum  is  lengthened  1  per  cent.     How  much 
does  it  lose  a  day? 

4.  A  pendulum  beating  seconds  at  one  place  is  carried  to  another 
station  where  it  gains  10  seconds   a  day.     Compare   the  values  of 
gravity  at  the  two  places. 

71.    The  Compound  or  Physical  Pendulum.  —  Let  the 
mass  of  a  heavj  pendulum,  including  its  suspension,  be  M, 
and   let   the  centre  of  inertia  of  this  mass  ' 
be    at   a  distance  h  from   the  axis    of  sus-  ' 
pension  A  (Fig.  41).     Then  the  total  force  - 
of  gravity,  or  the  weight  of  the  pendulum,  - 
is  Mg,  and  the  lever  arm  for  the  centre  of  . 
rotation  A  is  BM,  or  h  sin  0. 

The  moment  of  the  force  producing  rota- 
tion is  therefore 

Mgh  sin  6. 

The  moment  of  the  force  producing  rota- 
tion is  also  (Art.  66)  a^mr2. 

Hence  aZmr2  =  Mgh  sin  6. 

2mr2  __  g  sin  6 

~Mh       ~~aT 

The  second  member  of  this  equation  is  the  linear. tan- 
gential acceleration  divided  by  the  angular  acceleration. 


KINETICS.  95 

This  is  a  length  (65)  the  same  as  linear  tangential  velocity 
divided  by  angular  velocity.  The  equation  may  be  con- 
sidered then  as  a  general  formula  for  the  length  of  the 
ideal  simple  pendulum  which  will  oscillate  in  the  same 
time  as  the  real  compound  pendulum.  That_this  is  true 
may  be  demonstrated  by  supposing  the  mass  M  all  col- 
lected at  a  point  at  a  distance  I  from  A,  so  situated  that 
the  time  of  oscillation  as  a  simple  pendulum  is  the  same  as 
that  of  the  compound  physical  pendulum.  Then  the  r  for 
each  particle  becomes  /,  and  h  also  becomes  I  for  the  ag- 
gregated mass.  Hence 

ymr2  _      Iml2  _    Ml2  _  , 

^h     ~m  :  ~M 

The  length  I  is  called  the  length  of  the  equivalent  simple 
pendulum,  or  the  pendulum  vibrating  in  the  same  time  as 
the  actual  physical  pendulum.  It  is  found  by  dividing 
the  moment  of  inertia  of  the  pendulum  about  the  axis  of 
suspension  by  the  product  of  its  mass  and  the  distance 
between  its  centre  of  mass  and  the  axis  of  suspension. 

By  substitution  in  the  formula  of  the  last  article,  the 
period  becomes 

2'mr2 
Mgh' 

The  numerator  of  the  fraction  under  the  radical  sign  is 
the  moment  of  inertia  about  the  axis  around  which  the 
pendulum  must  oscillate,  and  the  denominator  is  the  stat- 
ical moment,  or  the  maximum  moment  of  the  couple 
producing  rotation,  when  the  lever  arm  is  h  or  when 

9  -i 

If  a  line  be  drawn  through  the  axis  of  suspension  A  and 
the  centre  of  mass  of  the  pendulum,  and  if  the  distance  I 
be  laid  off  on.  this  line  from  A,  its  lower  end  Avill  mark  the 


96 


MECHANICS. 


point  called  the  centre  of  oscillation.  It  is  of  course  the 
point  at  which  the  whole  mass  must  be  collected  so  as 
to  form  the  equivalent  simple  pendulum,  and  the  body  oscil- 
lates as  if  its  whole  mass  were  concentrated  there.  It  is 
also  called  the  centre  of  percussion,  because,  if  the  pen- 
dulum be  started  by  a  blow  at  this  point,  it  will  swing 
without  any  jar  on  its  supports.  A  bat  is  an  inverted 
pendulum,  and  drives  the  ball  best  without  jarring  the 
hands  if  it  strikes  the  ball  at  the  centre  of  percussion. 

72.    The  Reversibility  of  the  Pendulum.  —  Let  Fig.  42 
represent  a  physical   pendulum    consisting  of  a  uniform 
rectangle  or  bar.     Also  let  A  be  the  axis  of 
suspension,  G-  the  centre  of  mass,  and  0  the 
centre  of  oscillation. 

Applying  the  principle  of  the   moment    of 
inertia  about  a  parallel  axis  (Art.  69), 

l-h  ~lllh  '        Mh 

for  the  axis  through  A. 

Suppose  the  pendulum  to  be   reversed  and 
swung  from  an  axis  through  0,  and  let  I'  be 
Fig^42          the  length  of  the  equivalent  simple  pendulum 
for  this  axis.     Then 


G 


From  the  first  equation 


or 


Substitute  this  value  in  the  second  equation  and 


KINETICS.  97 

or  the  length  of  the  equivalent  simple  pendulum  is  the 
same  whether  the  pendulum  be  swung  from  A  or  from  0. 
If  the  length  is  the  same  the  time  of  oscillation  is  also  the 
same.  This  conclusion  is  of  course  independent  of  the 
form  -  of  the  compound  pendulum,  or  the  arrangement  of 
its  mass  about  the  axis. 

The  length  of  the  equivalent  simple  pendulum  is  then 
the  distance  between  the  two  axes  about  which  it  swings 
in  the  same  time. 

73.  Kater's  Reversible  Pendulum  (V.,  1,  238).  —  The 
reversible  pendulum  devised  by  Captain  Kater  utilizes  the 
principle  of  the  interchangeability  of  the  centres  of  sus- 
pension and  oscillation.  A  brass  bar,  carrying  at  one  of 
its  extremities  a  heavy  lens-shaped  weight,  has  fixed  in  it 
two  knife-edges  facing  each  other.  One  of  these  knife- 
edges  is  near  one  end  of  the  bar  and  the  other  is  next  to 
the  heavy  weight  at  the  other  end,  the  weight  being  out- 
side. Between  the  two  is  a  smaller  weight  adjustable  by 
means  of  a  tangent  screw.  This  weight  is  adjusted  till  the 
time  of  vibration  of  the  pendulum  is  the  same  whichever 
knife-edge  serves  as  the  axis  of  suspension.  The  length  I 
of  the  equivalent  simple  pendulum  is  then  rigorously 
known  and  is  equal  to  the  distance  between  the  two 
parallel  knife-edges. 

If  its  time  of  vibration  is  exactly  determined  by  the 
method  of  coincidences,  the  length  of  the  pendulum  giving 
seconds  in  air  can  be  determined.  The  mean  of  the  results 
of  Borda,  Biot,  and  Peirce  is 

I  =  993.92  mm. 

at  Paris  at  an  elevation  of  72  metres ;  this  is  for  a  vacuum 
and  is  doubtless  correct  to  within  y^j-th  of  a  millimetre. 
The  value  of  g  corresponding  to  this  length  is  980.96  at 
Paris. 


98 


MECHANICS. 


74.  Axis  for  Minimum  Period  of  Oscillation.  —  If 
the  pendulum  is  a  uniform  bar  or  rod,  then  for  the 
axis  of  suspension  at  one  end  A  (Fig.  43)  the  centre 
of  oscillation  is  at  0 ;  and  for  an  axis  at  the  other 
end  A',  the  centre  of  oscillation  is  at  0'.  The  time 
of  oscillation  about  these  four  axes  is  the  same.  If 
the  axis  passed  through  6r  the  time  of  oscillation 
would  be  infinite,  for  then  h  is  zero,  and  therefore  I 
is  infinite.  The  period  of  oscillation  decreases  as 
the  axis  is  shifted  from  6r  toward  0',  and  on  further 
shifting  to  A  the  time  is  the  same  as  at  0'.  If  the 
axis  could  be  further  removed  from  G~  the  period 
would  continue  to  increase.  It  must  therefore  pass 
through  a  minimum  between  0'  and  A. 
To  find  this  point  we  have  in  Article  72 


A 

Fig.  43. 


a  constant,  since  I0  is  the  moment  of  inertia  about  an 
axis  through  the  centre  of  mass,  perpendicular  to  the 
length  of  the  bar. 

Then  h  (I  —  h*)  is  a  constant. 

But  when  the  product  of  two  factors  is  a  constant  their 
sum  is  a  minimum  when  they  are  equal  to  each  other. 
Their  sum 

h  +  (I  -  A)  =  Z, 
and  I  is  therefore  least  when 


or  when  I  =  %h. 

The  condition  for  the  least  period  of  oscillation  is  then 
that  the  axis  of  suspension  and  the  centre  of  oscillation 
shall  be  equidistant  from  the  centre  of  mass. 

The  length  of  the  equivalent  simple  pendulum  for  this 
minimum  period  is  twice  the  distance  between  the  axis 
and  the  centre  of  mass. 


MECHANICS   OF  FLUIDS.  99 


CHAPTER   V. 

MECHANICS    OF    FLUIDS. 

75.  No  Statical  Friction  in  Fluids.  — The  term  fluid 
applies  both  to  liquids  and  gases.  A  perfect  fluid  would 
offer  no  resistance  to  a  shearing  stress  ;  but  owing  to  vis- 
cosity no  fluid  is  without  shearing  stress.  The  character- 
istic property  of  a  fluid  is  to  flow.  Viscosity  is  a  resistance 
to  flow  due  to  internal  friction  of  the  particles  of  the  fluid 
against  one  another.  A  fluid  possesses  no  rigidity,  but  is 
deformed  by  any  force,  however  small.  Ether,  water,  oil, 
molasses,  Canada  balsam,  sealing-wax,  shoemaker's  wax, 
are  examples  of  fluids  of  progressively  greater  and  greater 
viscosity.  Shoemaker's  wax,  which  acts  like  a  solid  when 
struck  Avith  a  blow,  deports  itself  like  a  liquid  because  the 
deformation  of  a  mass  of  it  continues  so  long  as  the  dis- 
torting force  continues  to  act  on  it.  It  accommodates 
itself  to  an  irregular,  sinuous  channel  and  flows  slowly 
down  it.  Corks  placed  under  a  layer  of  shoemaker's  wax 
on  water  gradually  come  through  the  wax  to  the  surface. 
Bullets  placed  on  top  of  the  wax  settle  through  it.  Masses 
of  wood  in  the  wax  slowly  settle  into  the  position  of 
stable  equilibrium  which  they  immediately  assume  in 
water. 

By  statical  friction  is  meant  the  friction  existing  between 
bodies  relatively  at  rest.  It  acts  tangential  to  the  surfaces 
in  contact.  The  absence  of  statical  friction  in  fluids  means 


100  MECHANICS. 

that  all  fluid  pressure,  when  the  fluid  is  at  rest,  is  normal 
to  the  surface  of  the  fluid.  If  this  pressure  were  oblique 
it  could  be  resolved  into  a  normal  and  a  tangential  com- 
ponent, and  the  latter  would  produce  motion  of  the  fluid. 
Since  action  and  reaction  are  opposite  in  direction  the  stress 
between  any  two  liquid  surfaces  in  contact,  or  between  a 
liquid  and  the  walls  of  the  containing  vessel,  must  be 
normal  to  both. 

When  fluids  are  in  motion  there  exists  a  tangential  force 
of  friction.  Hence  the  middle  of  a  stream  moves  faster 
than  the  sides.  A  glacier  moves  down  its  channel  in  the 
same  manner.  Tangential  friction  between  two  fluid  sur- 
faces is  due  to  viscosity. 

76.    Pascal's  Law In  discussing  the   conditions  -of 

equilibrium  of  liquids  it  is  often  useful  to  conceive  small 
portions  to  become  solidified  without  change  of  density,  or 
to  consider  such  portions  as  separately  recognizable  without 
altering  their  relations  to  the  surrounding  mass. 

Conceive  the  small  cube  (Fig.  44)  to 
become  solid  without  change  of  volume. 
Then  the  pressure  on  each  face  of  this 
cube  is  the  same,  whichever  way  it  is 
turned,  provided  the  fluid  is  at  rest,  and  is 
not  acted  on  by  any  forces  except  those 
applied  to  its  surface.  If  the  forces  on  any  pair  of  opposite 
faces  of  the  cube  were  unequal,  their  difference  would 
produce  motion  of  the  cube.  The  dimensions  of  the  cube 
may  become  those  of  a  material  particle  ;  then  the  pressure 
in  all  directions  at  the  point  must  be  the  same.  If  we 
consider  several  such  cubes  touching  one  another  along 
any  line,  the  pressure  between  them  throughout  the  line 
must  be  the  same  ;  for  there  is  no  difference  of  pressure 


MECHANICS   O:&  FLUID  A: 


101 


on  opposite  sides  of  any  one  cube,  and  any  difference  be- 
tween adjacent  cubes  in  opposite  directions  would  produce 
motion,  which  is  contrary  to  the  hypothesis  of  the  liquid 
at  rest.  Hence,  expressly  eliminating  the  influence  of 
gravity,  the  pressure  throughout  the  mass  of  a  liquid  at 
rest  is  everywhere  the  same.  This  is  known  as  Pascal's 
Principle. 

It  follows  that  an  increase  of  pressure  on  any  plane  is 
transmitted  to  every  other  point  in  the  fluid. 

Pressure  applied  to  any  area  of  a  confined  fluid  is  trans- 
mitted to  every  other  equal  area,  either  of  the  fluid  or  the 
walls  of  the  containing  chamber,  without  diminution. 
This  is  the  principle  of  the  transmission  of  pressure.  It 
is  the  principle  applied  in  the  Hydraulic  Press,  which  is 
employed  for  compression  purposes,  for  making  lead  pipe 


by  forcing  the  lead  through  a  die,  and  for  the  purpose  of 
working  cranes  to  lift  heavy  masses  of  metal,  and  the  like. 
It  consists  fundamentally  of  two  cylinders  of  different  size, 
aa  and  AA  (Fig.  45),  connected  by  a  pipe  CO.  The  small 
cylinder  aa  is  provided  with  an  inlet  and  an  outlet  valve, 
not  shown,  similar  to  those  of  a  force-pump.  The  plunger 


102 


in  a  is  worked  by  the  lever  L.  Water  is  thus  drawn  from 
the  surrounding  reservoir,  and  is  forced  through  the  pipe 
(7  (7  to  the  large  cylinder  F",  where  it  acts  on  -the  plunger  A. 
The  mechanical  advantage  of  the  machine  is  the  ratio  of 

the  cross-sections  of   the  plungers    a   and    A,   so    that   a 

j^ 
pressure  of  p  dynes  on  a  balances  p  —  dynes    on  A.     But 

what  is  gained  in  power  is  lost  in  speed,  for  the  plunger  a 
A 

must  travel  —  times  as  far  as  the  one  in  A.     The  work 

a 

done  on  a  is  thus  equal  to  that  done  by  A  if  friction  is 
neglected.  There  is  no  gain  of  energy,  but  the  liquid 
acts  as  a  practically  incompressible  medium  for  the  trans- 
mission of  pressure. 

77.  Pressure  the  same  at  all  Points  of  a  Horizontal 
Plane.  —  If  the  fluid  is  not  weightless,  Pascal's  Principle 
does  not  hold  good,  except  for  the  same  horizontal  plane. 
For  in  order  that  the.  cube  above  considered  shall  be  in 
equilibrium,  the  pressure  on  its  lower  surface  must  exceed 
the  pressure  on  its  upper  surface  by  the  weight  of  the  cube 
itself.  But  the  pressures  at  all  points  in  a  horizontal  plane 
still  remain  equal  to  one  another.  For  consider  a  small 
cylinder  of  the  fluid  with  its  axis  horizontal  and  its  ends 
minute  vertical  planes.  Then  since  the  cylinder  is  at  rest 
the  pressures  on  its  ends  are  equal  horizontal  forces.  But 
the  cylinder  may  be  in  any  position  in  the  plane.  The 
horizontal  forces  throughout  the  plane  are  then  everywhere 
the  same.  Moreover,  the  application  of  the  reasoning 
employed  in  the  last  article  to  any  minute  cube  in  the 
plane  would  show  that  the  forces  acting  on  the  cube  re- 
duced to  a  material  particle  are  the  same  in  all  directions. 
This  holds  true  for  all  the  particles  throughout  the  plane  ; 
so  that  the  forces  at  all  points  in  the  same  horizontal  plane 


MECHANICS    OF  FLUIDS. 


103 


are  the  same  in  all  directions.  But  while  this  is  true  for 
every  horizontal  plane,  the  value  of  the  force  changes  from 
plane  to  plane  because  of  the  weight  of  the  liquid. 

78.  The  Free  Surface  of  a  Liquid  Horizontal Con- 
sider a  particle  m  at  some 
point  B  (Fig.  46)  of  the  free 
surface  ABD,  which  we  may 
suppose  is  not  horizontal. 
The  vertical  force  on  m  is 
mg,  represented  by  B  W.  Re- 
solve this  into  two  compo- 
nents, one  normal  and  the 
other  tangential  to  the  sur- 
face at '  B.  The  latter  component  BC  will  cause  the 
particle  to  move.  But  by  hypothesis  the  liquid  is  at  rest. 
Hence  there  can  be  no  tangential  component.  But  this 
tangential  component  disappears  only  when 
the  surface  is  horizontal.  Therefore,  the  free 
surface  of  a  liquid  is  a  horizontal  surface,  and 
the  lines  of  force  due  to  gravity  are  every- 
where normal  to  it. 


Fig.  46. 


79.  Pressure  Varies  directly  as  the 
Depth.  —  The  pressure  at  every  point  in  an 
incompressible  fluid  due  to  its  weight  is  pro- 
portional to  the  depth  beneath  the  surface. 
For  consider  two  vertical  cylinders  of  liquid 
with  their  upper  ends  in  the  surface  of  the 
liquid  at  rest  (Fig.  47).  Let  the  length  of  a  be  n  times 
that  of  b.  Then  the  weight  of  the  one  is  n  times  as  great 
as  that  of  the  other,  and  equilibrium  can  exist  only  when 
the  upward  pressure  on  the  base  of  a  is  n  times  as  great 


Fig.  47. 


104 


MECHANICS. 


as  on  that  of  b;  for  the  upward  pressure  'is  balanced  by 
the  weight  of  the  column  of  liquid.  The  pressure  on  the 
plane  at  a  depth  n  is  therefore  n  times  as  great  as  at  a 
depth  unity,  or  the  pressures  are  proportional  to  depths. 

80.  Two   Liquids   in   Communicating   Tubes.  --  Let 

two  liquids  of  different  density  be  placed 
in  the  two  limbs  of  a  bent  tube  (Fig.  48). 
Then  their  heights  above  the  horizontal 
plane  drawn  through  their  surface  of  sepa- 
ration are  inversely  as  their  densities.  For 
let  their  heights  be  h  and  li'  and  their  den- 
sities d  and  d'  respectively.  Then  since 
the  pressure  is  everywhere  the  same  on  the 
horizontal  plane  of  separation,  the  press- 
ures due  to  the  two  columns  of  heights 
^  anc^  h'  mnst  also  be  the  same.  But  the 
volumes  for  a  column  of  unit  cross-section 
are  h  and  h',  and  volumes  multiplied  by 
densities  give  masses.  Hence 
=  h'd'g, 

h      d' 
—  —  —  . 

JF     d 

The  heights  are  therefore  inversely 
as  the  densities. 

81.  Back  Pressure  on  a  Discharg- 
ing Vessel.  -  -  The  vertical  sides  of 
a  vessel  containing  a  liquid  sustain 
equal  lateral  pressures  on  the  same   - 
level  so  long  as  the  liquid  is  at  rest. 

The  pressures  acting  outward  over  the  whole  surface  have 
therefore  a  resultant  horizontally  equal  to  zero.  But  let 
an  opening  be  made  at  any  point  A  (_Fig.  4i))  ;  the  pressure. 


Fig  48 


or 


MECHANICS    OF  FLUIDS.  105 

inward  at  that  point  is  removed  by  removing  the  reacting 
wall,  while  the  outward  pressure  on  the  opposite  wall  re- 
mains uncompensated.  Hence  the  entire  vessel  tends  to 
move  in  the  direction  opposite  to  that  of  the  stream.  This 
is  the  principle  of  Barker's  mill,  of  rockets,  and  of  rotat- 
ing fireworks. 

It  is  perhaps  better  to  apply  to  the  discharging  vessel 
the  third  law  of  motion.  The  momentum  of  the  stream  in 
one  direction  is  equal  to  the  stress  on  the  vessel  in  the 
other,  or  the  action  and  the  reaction  are  equal  to  each  other. 
The  same  principle  may  be  illustrated  with  air  pressure  by 
means  of  a  contrivance  similar  to  a  Barker's  mill  screwed 
on  the  plate  of  an  air-pump  and  covered  by  a  receiver. 
The  air  is  first  exhausted.  If  it  is  then  admitted  at  at- 
mospheric pressure  by  means  of  the  admit  cock,  the  appa- 
ratus spins  around  with  great  rapidity  on  account  of  the 
uncompensated  back  pressure,  or  the  reaction  of  the  issuing 
jets  of  air  on  the  walls  of  the  tubes. 

82.  Total  Pressure  on  any  Immersed  Surface.  —  Let 

a  (Fig.  50)  be  any  very  small  surface,  the  distance  of 
which  from  the  free  surface  of  the  liquid 
is  li.  Since  it  is  very  small,  the  pressure 
sustained  by  it  is  independent  of  its 
position  relative  to  the  horizontal  plane 
through  it.  This  pressure  is  ahdg,  in 
which  d  is  the  density  of  the  liquid. 
The  height  h  must  be  ia  cms.,  and  a  must 

l)e  measured  in  square  cms.     The  pressure  will  then  be  in 

dynes. 

Let  the  area  A  be  considered   as    composed  of  a  large 

number  of  small  elements,  so  that 

A    =    di   +    a2  +    a3  +          '         '          • 


106  MECHANICS. 

Then  the  pressures  on  these  areas  will  be 

a\h\dy,  <tJi.2<lf/-,  ";;/':{' 7//,  etc., 
and  the  total  pressure  on  A, 

P  —  (ai^j  +  a27i2  +  ajis  +      ...)%. 

But  by  Art.  62  the  quantities  within   the  parenthesis 
may  be  put  equal  to  AH,  where  H  is  the   depth   of  the 
centre'  of  inertia  of  the  entire  surface  A,  the  plane  of  refer- 
ence being  the  surface  of  the  liquid.     Therefore 
P  =  AHdg. 

This  expression  is  the  weight  of  a  prismatic  or  cylindrical 
column  of  the  liquid,  the  base  being  the  immersed  surface 
A,  and  the  height  equal  to  the  distance  of  its  centre  of 
inertia  below  the  free  surface  of  the  liquid.  This  is  of 
course  the  pressure  on  one  side  of  the  immersed  surface 
only.  It  applies  equally  well  to  the  estimation  of  the  press- 
ure on  any  portion  of  the  walls  of  the  vessel  or  to  retaining 
walls  in  general. 

PROBLEMS. 


x 


1.  Let  a  hollow  cube  be  filled  with  water.     The  pressure  on  the 
sides  is  how  many  times  the  pressure  on  the  bottom  ? 

2.  A  spherical  shell  of  radius  r  cms.  is  filled  with  water.     Esti- 
mate the  total  hydrostatic  pressure  on  the  interior.   A"    X 

3.  A  retaining  wall  2  metres  wide  and  50  metres  long  is  inclined 
at  an  angle  of  30°  with  the  vertical.     Find  the  total  pressure  of  water 
in  kilogrammes  against  it  when  the  water  rises  to  the  top.     Find  also 
the  horizontal  pressure  against  it. 

4.  Find  the  pressure  in  grammes  on  the  bottom  and  sides  of  a 
cubical  vessel,  10  cms.  on  each  side,  filled  with  mercury.     The  den- 
sity of  mercury  is  13.596.  \ 

X*^ 

83.  The  Centre  of  Pressure The  centre  of  hydro- 
static pressure  on  any  immersed  surface  is  the  point  of 
application  of  the  resultant  of  all  the  elementary  hydro- 


MECHANICS    OF   FLUIDS. 


107 


0 


G 


static  pressures  against  the  elements  of  the  surface.  If  the 
area  is  plane  all  these  elementary  pressures  are  parallel, 
and  the  problem  consists  in  finding  the  resultant  of  a  sys- 
tem of  parallel  forces.  This  problem  we  can  treat  here 
only  in  an  elementary  manner  by  means  of  a  couple  of 
examples.  Thus,  to  find  the  centre  of  pressure  on  a  rec- 
tangle with  one  side  in  the  liquid  surface,  divide  the 
rectangle  (Fig.  51)  into  a 
very  large  number  n  of  equal  ^^ 
areas  by  lines  parallel  to  the 
surface.  Since  the  areas  are 
equal  the  pressures  on  them 
are  simply  proportional  to  the 
depth,  and  the  centre  of  press- 
ure of  each  small  area  is  on  the  line  0(7,  which  bisects 
all  of  them.  Hence  the  centre  of  pressure  of  the  rectangle 
is  on  this  same  line.  Let  it  be  at  G.  To  find  the  dis- 
tance OGr. 

The  pressures  on  the  several  areas  will  be  a  constant 
multiplied  by  the  integers  1,  2,  3,  etc.  This  constant  in- 
cludes the  area  of  each  strip,  the  density  of  the  liquid,  and 
the  cosine  of  the  angle  which  the  rectangle  makes  with  a 
vertical  plane. 

Then  the  total  pressure  is 


Fig.  51. 


c  (1  +  2  +  3  + 


=  4= 


Then  applying  the  general  principle  of  Art.  62 


c  (I2  +  22  +  32  + 


2\  W  V 

n2)  =  c—  =  2px. 


TT  TT      s\si  . 

Hence      X=  Oa  =          ,=  c^  ^  tf_  =_  . 

But  n  is    the  number   of    equal  divisions    or  the  total 


108  MECHANICS. 

depth  of  the  rectangle.     Hence  the  centre  of  pressure  is 
f  the  depth  of  the  rectangle. 

84.  Centre  of  Pressure  on  an  Immersed  Triangle.  — 
Let  the  apex  be  at  the  surface 
and  the  base  horizontal.  Divide 
the  triangle  (Fig.  52)  into  a  large 
number  of  sections  n  by  equidis- 
tant lines  parallel  to  the  surface. 
Then  the  areas  of  these  sections 
will  be  proportional  to  the  depths 

Fig- 52.  10Q 

1,  2,  3,     .     .     .     n. 

The  pressures  upon  them  are  proportional  jointly  to  the 
areas  and  to  their  depths  below  the  surface.  We  may 
therefore  write  the  total  pressure  in  this  case 

c  (Y-  +  22  +  32  + n*)  =  4*  -  2p. 

o 

Draw  a  line  from  0  to  the  middle  point  of  the  base. 
It  will  bisect  all  the  small  areas,  and  will  therefore  pass 
through  the  centre  of  pressure  of  each  one  of  them.  The 
centre  of  pressure  on  the  triangle  therefore  lies  on  this 
line.  Let  it  be  at  6r,  at  a  distance  X  from  0.  Then  the 
products  px  may  be  written 

c  (I3  +  23  +  33  +      .     .     .     .     n3)  =  c^    =  IpX. 

Hence          X  =  Oa  =  2[pX  =  c  f  ~  /'      ~  n. 

i>         4         3       4 

But  n  is  the  number  of  equal  divisions  of  the  line  from 
0  to  the  middle  point  of  the  base.  Hence  the  centre  of 
pressure  is  found  by  measuring  three-fourths  the  depth 
from  the  apex  to  the  middle  point  of  the  base. 

In  a  similar  way,  if  the  base  were  in  the  surface  of  the 


MECHANICS    OF   FLUIDS. 


109 


liquid,  it  can  be  shown  that  the  centre  of  pressure  would 
be  half  way  down  from  the  middle  point  of  the  base  to  the 
apex. 

PROBLEM. 

A  dam  whose  section  is  a  right-angled  triangle,  3  metres  high,  is 
built  of  stone  of  density  3.  If  the  water  reaches  the  top  on  the 
vertical  side,  what  must  be  the  breadth  of  the  base  with  a  factor  of 
safety  of  ten,  assuming  that  the  wall  may  be  treated  as  a  rigid  body? 

85.  The  Principle  of  Archimedes. — When  a  body  is 
immersed  in  any  fluid  it  apparently  loses  weight.  It  is  in 
reality  partly  supported  by  the  fluid,  or  is  subjected  to  an 
upward  pressure  equal  to  the  weight  of  the  fluid  displaced. 
This  follows  at  once  if  we  consider  that  the  body  has  re- 
placed an  equal  volume  of  the  fluid  itself  which  was  kept 
in  equilibrium  by  an  upward  pressure  equal  to  its  own 
weight.  The  upward  pressure  on  the  immersed  body  is 
the  same  as  that  on  the  fluid  which  it  replaces. 

"Let  a  cube  of  any  heavy  material  be  immersed  in  water 
(Fig.  53).     The  opposite  lateral   faces  a   and  b  will  be 
equally  pressed  in  opposite  directions.     The  same  will  be 
true    of   the  other  pair  'of  lateral 
faces.     On  d  there  will  be  a  down- 
ward pressure  equal  to  the  column 
of  water  with  base  d  and  height 
nd.     On  the  bottom  c  there  will  be 
an  upward  pressure  equal  to   the 
weight   of  the   column   of   water, 
whose  base  is  c  and  height  nc. 

The  resultant  upward  pressure 
on  the  solid  is  the  difference  of  the 
pressures  on  the  bottom  and  top 

of  the  cube,  and  this  difference  is  the  weight  of  the  cube 
of  water  of  the  same  dimensions  as  the  solid. 


110  MECHANICS. 

Since  the  principle  applies  to  gases  as  well  as  liquids,  it 
may  be  stated  generally  as  follows :  A  body  immersed  in  a 
fluid  is  buoyed  up  by  a  force  equal  to  the  weight  of  the 
fluid  displaced  by  it. 

The  centre  of  buoyancy  is  the  name  applied  to  the  centre 
of  mass  of  the  displaced  fluid. 

When  a  body  floats  in  a  liquid  it  sinks  to  such  a  depth 
that  the  weight  of  the  liquid  displaced  just  equals  its  own 
weight.  If  the  weight  of  the  body  is  more  than  the 
weight  of  the  liquid  which  it  can  displace,  it  will  then 
sink;  if  less  it  will  float.  Thus  a  solid  iron  ball  sinks  in 
water,  since  its  density  is  7.8,  or  its  own  weight  is  7.8 
times  as  great  as  the  water  displaced  by  it.  If,  however, 
it  is  made  in  the  form  of  a  hollow  ball,  it  may  displace 
a  mass  of  water  equal  in  weight  to  its  own. 

On  the  contrary,  iron  cannot  be  made  to  sink  in  mer- 
cury,, because  the  density  of  mercury  is  13.596.  Iron, 
therefore,  floats  on  liquid  mercury. 

86.  Density  and  Specific  Gravity.  -  -  Density  has 
already  been  defined  as  the  mass  in  grammes  contained  in 
a  cubic  centimetre  of  volume. 

Specific  gravity  is  the  ratio  of  the  density  of  the  body  to 
that  of  another  body  taken  as  a  standard.  If  »  denotes 
the  specific  gravity  of  a  body,  d  its  density,  and  D  the 
density  of  the  standard,  then 

d  =  sD, 
and  M=  VsD. 

If  the  density  of  the  standard  is  unity,  densities  and 
specific  gravities  are  numerically  equal  to  each  other.  In 
the  metric  system  the  mass  of  a  cubic  centimetre  of  dis- 
tilled water  at  4°  C.  is  one  gramme,  or  the  density  of  water 
is  unity  under  standard  conditipns.  The  density  and  the 


MECHANICS    OF  FLUIDS.  Ill 

specific  gravity  of  any  body  in  this  system  are,  therefore, 
numerically  identical. 

If,  however,  the  English  gravitational  system  is  employed 
then  the  density  of  water  is  about  62.4,  since  a  cubic  foot 
of  distilled  water  at  4°  C.  contains  a  mass  of  62.4  Ibs. 

87.  Specific  Gravity  of  Solids.  —  a.     Bodies  heavier 
than  water.     The   density  or   specific   gravity  of  a   solid 
insoluble   in  water  may  be  found  by  weighing  the  body 
first  in  air  and  then  suspended  in  water.     Its  apparent 
loss  of  weight  in  water  is,  by  the  principle  of  Archimedes, 
the  weight  of  the  water  displaced.     Hence  the  quotient 
of  the  weight  in  air  by  the  loss  of  weight  in  water  is  the 
density.    If  the  water  is  at  a  temperature  above  (or  below) 
the  maximum,  then  the  value  found  by  the  process  just 
described  must  be  multiplied  by  the  density  D  of  the  water 
at  the  temperature  at  which  the  observation  was  made,  or 

d  =  sD. 

b.  Bodies  lighter  than  water.  Employ  a  sinker  suffi- 
cient to  make  the  body  sink  in  water.  Counterbalance 
with  the  body  in  the  scale  pan  and  the  sinker  suspended 
from  the  pan  and  immersed  in  the  water.  Transfer  the 
body  from  the  scale  pan  to  the  sinker  in  the  water.  The 
weight  w'i  which  must  be  added  to  the  scale  pan  to  re- 
store the  equilibrium,  is  the  weight  of  the  water  displaced. 
It  is  not  necessary  to  know  the  weight  of  the  sinker. 
Then  if  w  is  the  weight  of  the  body  in  air,  the  ap- 

w 
parent  density  is  — . 

88.  Density  of  Liquids  inferred  from  Loss  of  "Weight. 
-  The  density  of  liquids  may  be  determined  by  means  of 

the  principle  of  Archimedes.     A  glass  sinker  is  weighed 
in   air,     and   then    the    loss  of   weight    it   sustains  when 


112  MECHANICS. 


immersed  in  water  and  in  the  liquid  under  examination  is 
determined.  Let  these  apparent  losses  be  w  and  w1.  Then 

the  apparent  density  of  the  liquid  is  — ,  since  its  apparent 

loss  of  weight  in  the  two  cases  is  the  weight  of  the  same 
volume  of  the  two  liquids.  If  the  water  is  not  at  4°  0. 
the  result  must  be  corrected  as  before. 

89.  General  Theory  of  Hydrometers  of  Variable  Im- 
mersion. -  -  The  approximate  density  of  liquids  may  be 
conveniently  determined  by  means  of  an  hydrometer, 
which  consists  of  a  straight  stem  of  glass,  with  a  bulb  at 
the  bottom,  and  weighted  so  as' to  float  to  the  proper  depth 
in  a  vertical  position.  The  graduation  of  hydrometers 
must  be  made  experimentally.  Those  with  equidistant 
divisions  on  the  stem  have  their  constants  determined  as 
follows :  Let  v  be  the  volume  immersed,  in  units  of  the 
divisions  of  the  stem,  when  the  hydrometer  sinks  to  the  zero 
of  the  scale.  Then  if  d  and  dl  are  the  densities  of  two 
liquids,  and  n,  %  (Fig.  53a)  the  corresponding  divis- 
ions of  the  scale  to  which  the  instrument  sinks  in 
them,  the  zero  being  near  the  top,  the  volumes  im- 
~n  mersed  in  the  two  cases  are  v  —  n  and  v  —  %. 

Hence  (v  —  n)  d  =  (v  —  %)  d, , 

since  the  masses  of  the  liquids  displaced  are  both 
^/  equal  to  the  mass  of  the  hydrometer.     Therefore 

d  -  t/r ' 

If  one  of  the  liquids  is  water  and  the  hydrometer 
sinks  in  it  to  zero,  then  n  =  0  and  d  —  1. 
U  In  that  case 

n^di 

Fig.  53a.  V  =  —        — . 


MECHANICS    OF  FLUIDS.  118 

For  any  other  liquid  of  density  D,  in  which  the  hydrom- 
eter sinks  to  division  JV,  the  equation  of  equilibrium  is 

D  (v  _  jV)  =  v,  or  D  = 


v  —N 

Since  the  density  of  water  is  unity  its  mass-  is  numeri- 
cally equal  to  its  volume.  The  volume  having  been 
determined,  a  table  can  be  made  giving  the  densities 
corresponding  to  the  various  divisions  of  the  scale. 

9O.  Baume's  Hydrometer  ---  The  two  liquids  are  water 
and  salt  water,  containing  15  per  cent  of  salt,  of  density 
1.11383  at  17°.8  C.  The  division  to  which  the  hydrometer 
sinks  in  the  salt  water  is  marked  15. 

»-      - 

n  146.78 

and  I)  = 


U6.1S-N 

This  formula  gives  the  density  corresponding  to  any 
division  JV. 

For  liquids  lighter  than  water  the  zero  of  the  scale 
is  placed  near  the  bottom  of  the  straight  stem.  It  is 
placed  at  the  point  to  which  the  instrument  sinks  in  a  10 
per  cent  salt  solution;  the  point  to  which  it  sinks  in 
water  is  marked  10. 

A  general  formula  is  obtained  as  before  by  reversing  the 
signs  of  n  and  w19  and  makings  =  10  for  water.  Then 

Mi  — 10 

v  =-^—7— ; 

1—6?! 

and  if  d^  is  the  density  of  the  10  per  cent  salt  solution  ior 
which  MI  =  0,  then 

10 


114  MECHANICS. 

For  other  liquids 


v  +  N 

Gerlach  obtained  for  v,  135.88.     Hence 
J)=      145.88 


135.88  + 


91.  Fundamental  Phenomena  of  Capillary  Action.  — 
Capillary  action  consists  in  tlie  elevation  or  depression  of 
liquids  along  the  walls  of  the  vessel  containing  them,  in  the 
ascent  or  depression  of  liquids  between  two  plates  very 
close,  together,  or  in  tubes  of  such  small  inner  diameter 
that  they  approach  the  dimensions  of  a  hair  ;  whence  the 
name  capillarity,  from  capillu*,  a  hair. 

It  is  easy  to  determine  that  the  free  surface  of  a  liquid 
is  not  horizontal  near  the  sides  of  the  vessel  containing  it, 
but  presents  a  noticeable  curvature.  When  the  liquid 
wets  the  vessel,  as  water  in  glass,  the  surface  is  concave, 
or  the  water  rises  along  the  glass  ;  on  the  other  hand, 
when  the  liquid  does  not  adhere,  as  in  the  case  of  mercury 
and  glass,  the  surface  is  convex. 

With  the  former  conditions  when  small  tubes,  less  than 

two  millimetres  in  diameter, 
are  supported  in  the  liquid, 
the  liquid  is  perceptibly 
higher  in  the  tube  than  the 
level  surface  without;  with 
the  conditions  determining 
a  convex  surface  the  level 
of  the  liquid  within  the 

tube  is  below  that  outside  (Fig.  54).     This  elevation  or 
depression  of  the  liquid  is  inversely  as  the  diameter  of  the 


MECHANICS   OF  FLUIDS.  115 

tube,  provided  this  diameter  does  not  exceed  two  milli- 
metres. 

The  phenomena  are  independent  of  the  pressure  to 
which  the  liquid  is  subjected. 

They  do  not  depend  upon  the  thickness  of  the  tube,  or 
in  other  words  the  action  between  the  liquid  and  the  tube 
is  limited  to  insensible  distances. 

The  elevation  or  depression  varies  with  the  jnateriaLof 
the  tube  and  the  nature  of  the-Jiqwidr.  The  elevation  of 
water  in  glass  is  greater  than  that  of  any  other  liquid, 
being  nearly  three  times  as  great  as  for  sulphuric  ether 
and  bisulphide  of  carbon. 

Both  the  elevation  and  depression  decrease  with  rise  of 
temperature.  An  elevation  of  water  amounting  to  132 
mm.  at  0°  C.  is  reduced  to  106  mm.  at  100°  C. 

Capillary  action  explains  the  diffusion  of  liquids  of  slight 
viscosity  through  porous  septa,  as  well  as  their  absorption 
by  porous  bodies.  Liquids  which  thus  pass  through  porous 
diaphragms  are  called  crystalloids;  while  glutinous  solu- 
tions of  gum,  starch,  albumen,  and  the  like,  which  do 
not  pass  through  porous  septa,  are  called  colloids.  They 
are  viscous,  diffuse  slowly,  and  are  indisposed  to  crystallize. 
Physiologists  attach  great  importance  to  this  distinction, 
inasmuch  as  it  explains  the  interchange  of  liquids  which 
goes  on  through  the  tissues  and  vessels  of  the  animal  sys- 
tem, as  well  as  the  absorption  of  water  by  the  spongioles 
of  roots. 

Joseph  Henry  concluded  that  mercury  passes  through 
lead  by  capillary  action ; J  also  that  silver  may  penetrate 
into  the  pores  of  copper  when  heated. 

92.  Law  of  Force  assumed  (A,  and  B.,  90) The 

attraction  of  gravitation  between  masses  of  matter  is  much 

1  Scientific  Writings,  Vol.  I.,  pp.  146,  228. 


116  MECHANICS, 

too  small  to  account  for  capillary  phenomena.  But  they 
can  be  explained  if  we  assume  an  attraction  between  the 
molecules.  The  total  expression  for  the  stress  between 
two  molecules  m  and  m'  then  becomes 


F=C  —  +  mm'f(f). 

The  first  term  expresses  the  attraction  of  gravitation ; 
the  second  is  the  molecular  attraction  giving  rise  to  capil- 
lary phenomena.  All  that  is  known  about  this  function  of 
r  is  that  it  is  very  large  for  insensible  distances,  that  it 
diminishes  very  rapidly  as  r  increases,  and  that  it  vanishes 
while  r  is  still  very  small.  The  maximum  value  of  r  at 
which  molecular  action  ceases,  called  e,  is  estimated  by 
Quincke  to  be  0.000005  cm.  or  0.000002  inch.  It  is  called 
the  range  or  radius  of  molecular  action.  Within  this 
distance  the  first  term  in  the  expression  for  the  stress  is 
quite  negligible  in  comparison  with  the  second. 

The  constant  C  in  gravitation  has  a  value  of  about 
1/40002,  that  is,  a  mass  of  nearly  4000  gms.  placed  at  a  dis- 
tance of  one  centimetre  from  an  equal  mass  would  attract 
it  with  a  force  of  one  dyne. 

93.  Surface  Tension  (A.  and  B.,  91;  B.,  2O1 ;  Tait's 
Properties  of  Matter,  227).  — If  the  molecules  of  a  liquid 
are  in  equilibrium,  then  the  conditions  of  the  molecular 
balance  in  the  interior  of  the  liquid  are  different  from 
those  at  the  surface.  At  any  point  in  the  interior  of 
the  liquid,  at  a  distance  from  the  surface  greater  than  €, 
each  molecule  is  attracted  equally  in  all  directions.  But 
near  or  at  the  surface  the  attraction  downward  is  not 
balanced  by  an  equal  attraction  upward,  and  the  molecules 
along  the  surface  are  therefore  packed  together  so  as  to 
compose  an  enveloping  film  of  thickness  e. 


MECHANICS    OF  FLUIDS. 


117 


m 


Fig.  55. 


Consider  a  liquid  bounded  by  a  plane  surface  mn  (Fig. 
55),  and  let  m'n'  be  a  parallel  plane  at  a  distance  e  below 
the  surface. 

If  we  imagine 
any  plane  passed 
through  a  point  in 
the  mass  of  the  liq- 
uid below  m'n1,  the 
normal  pressure  on 
this  plane,  due  to 
molecular  action,  is 
independent  of  the  direction  of  the  plane  with  respect  to 
the  surface  ;  for  the  number  of  molecules  acting  on  the 
point  is  the  same  in  every  direction.  If,  however,  the  point 
is  at  P,  nearer  the  surface  than  m'n1,  about  P  as  a  centre 
and  with  a  radius  e  describe  a  sphere.  Then  the  normal 
pressure  on  the  plane  through  P  perpendicular  to  m'n1  is 
greater  than  when  the  plane  is  parallel  to  m'n1.;  for  the 
upward  attraction  on  the  point  varies  from  a  maximum  at 
m'n'  to  zero  at  the  surface,  since  as  P  rises  the  upper  half 
of  the  sphere  described  about  P  contains  a  diminishing 
number  of  molecules.  From  this  inequality  of  pressure 
there  results  a  stress  or  tension  which  causes  the  surface 
to  contract;  and  this  tendency  to  contract  means  that  the 
surface  acts  like  a  stretched  membrane. 

If  the  surface  be  enlarged  by  forcing  molecules  out  along 
the  plane  through  P  normal  to  the  surface,  then  work 
must  be  done  upon  them  to  transfer  them  from  the  in- 
terior against  the  force  pressing  the  molecules  together 
along  the  surface.  In  other  words,  an  increase  in  the  area 
of  the  surface  means  an  increase  in  the  potential  energy  of 
the  liquid.  But  potential  energy  always  tends  to  become 


118  MECHANICS. 

a  minimum.  The  surface,  therefore,  contracts  to  as  small 
dimensions  as  the  conditions  allow. 

The  volume  enclosed  by  a  sphere  is  a  maximum,  or  the 
surface  itself  is  as  small  as  possible.  A  mass  of  free 
liquid  always  tends  therefore  to  assume  the  spherical  form 
except  as  it  is  distorted  by  other  forces.  Drops  of  rain 
and  dew  are  spherical  because  of  surface  tension  and  not 
because  of  gravity.  So  also  when  a  stream  of  molten  lead 
flows  from  a  small  orifice,  the  surface  tension  causes  the 
detached  masses  to  form  into  spheres  as  the  stream  breaks. 
If  they  rotate  as  they  descend,  they  remain  quite  spherical 
and  strike  the  water  at  the  bottom  of  the  shot-tower  as  shot. 

An  ingenious  method  of  separating  the  perfect  shot 
from  the  imperfect  ones  consists  in  causing  all  together  to 
roll  down  a  smooth  inclined  plane.  Near  the  bottom  is  a 
transverse  slit  in  the  plane.  The  perfect  shot  acquire 
enough  momentum  to  carry  them  safely  across,  while  the 
imperfect  ones  hobble  along  and  fall  into  the  crevasse. 

The  tendency  of  liquid  masses  to  assume  a  spherical 
form  is  best  illustrated  by  means  of  oil  relieved  from  the 
effect  of  gravity  by  immersion  in  a  liquid  of  its  own 
density. 

A  mixture  of  alcohol  and  water  is  made  of  the  same 
density  as  olive  oil.  Masses  of  olive  oil  placed  in  this 
liquid  will  neither  rise  nor  sink,  but  will  assume  a  globular 
form.  If  the  limiting  conditions  imposed  upon  them  do 
not  permit  them  to  assume  the  globular  form,  they  will  in 
every  case  assume  interesting  geometrical  forms  having 
the  smallest  superficial  area  under  the  given  conditions. 
If,  for  example,  a  circular  iron  ring  be  immersed  in  a  large 
mass  of  the  floating  oil,  and  some  of  the  oil  be  then 
removed  by  means  of  a  pipette,  the  remaining  mass  will 
take  the  form  of  a  double  convex  lens. 


MECHANICS   OF  FLUIDS. 


119 


94.  Further  Illustrations  of  Surface  Tension.  —  Float 
two  bits  of  wood  on  water  parallel  to  each  other,  and  a 
few  millimetres  apart.  Let  a  drop  of  alcohol  fall  011  the 
water  between  them,  and  they  will  suddenly  fly  apart. 
The  surface  tension  of  alcohol  is  less  than  that  of  water. 
The  effect  of  the  alcohol  is  therefore  to  weaken  the  film 
between  the  bits  of  wood.  The  parts  of  the  film  are 
thereby  separated  and  carry  the  wood  with  them. 

Place  a  thin  layer  of  water  on  a  piece  of  clean  glass, 
and  let  a  small  drop  of  colored  alcohol  fall  on  it. 

The  weak  spot  made  by  the  alcohol  causes  the  film  to 
break,  while  the  tension  about  it  draws  the  water  away, 
leaving  the  alcohol  surrounded  by  a  dry  area. 

Make  a  ring  of  stout  wire  three  or  four  inches  in 
diameter  (Fig.  56),  with  a  handle.  Tie  to  this  a  loop  of 
thread  so  that  the 
loop  may  hang  near 
the  middle  of  the 
ring.  Dip  the  ring 
into  a  good  soap  solu- 
tion containing  glyc- 
erine, and  obtain  a 
plane  film.  The 
thread  will  float  in 
it.  Break  the  film 
inside  the  loop  with 
a  warm  pointed  wire, 
and  the  loop  will  spring  out  into  a  circle.  The  ten- 
sion of  the  film  attached  to  the  thread  pulls  it  out  equally 
in  all  directions.  By  tilting  the  ring  from  side  to  side  the 
circle  may  be  made  to  float  about  on  the  film. 

A  small  bit  of  camphor  gum  placed  on  warm  water, 
perfectly  free  from  any  oily  film,  will  execute  rapid  and 


Fig.  56. 


120  MECHANICS. 

irregular  gyrations  and  movements  across  the  surface.  The 
camphor  dissolves  unequally  at  different  points,  and  thus 
produces  an  unequal  weakening  of  the  surface  tension  of 
the  water  in  different  directions. 


95.    Energy  and  Surface  Tension  (A.  and  B.,  93).  - 
If  we  call  the  loss  of  potential  energy,  due  to  a  diminution 
in  the  surface    of  one  unit,  the    surface    energy   per   unit 
area,  it  can  be  shown  that  this  is  numerically  equal  to  the 
surface  tension  for  unit  width  of  the  film.    Let  a  liquid  film 

be   stretched   on    a   frame  BCD 
C  with  the   light   rod   A   movable 

(Fig.  57).  Let  the  length  of  the 
D  rod  to  which  the  film  is  attached, 
that  is,  the  distance  between  B 
and  D,  be  «,  and  let  the  rod  be 
drawn  toward  C  a  distance  b. 


B 


a 


A 
Fi    57  Then  the  diminution  in  surface 

is  ab  units ;  and  if  E  is  the 

surface  energy,  the  potential  energy  has  decreased  by  an 
amount  equal  to  Eab. 

Further,  if  T  is  the  surface  tension  per  unit  width  of 

the  film,  the  total  surface  tension  lifting  the  rod  is   Ta. 

The  distance  moved  is  b.     Hence  the  work  done  against 

gravity  is  Tab.    This  equals  the  loss  in  potential  energy,  or 

Tab  =  Eab. 

Therefore  T=  E,  or  the  surface  energy  per  unit  area  is 
equal  to  the  surface  tension  per  unit  width. 

If  both  sides  of  the  film  are  taken  into  account  the  re- 
sult is  the  same. 

For  a  soap-film  in  air  the  surface  tension  is  27.45  dynes 
per  centimetre  width.  Hence  the  surface  energy  is 
27.45  ergs  per  square  centimetre.  For  pure  water  and  air 
the  surface  energy  is  81  ergs  per  square  centimetre. 


MECHANICS   OF  FLUIDS. 


121 


96.  Capillary  Elevation  or  Depression  explained  by 
Surface  Tension.  —  Let  h  (Fig.  58)  be  the  mean  elevation 
of  the  liquid  in  the  tube 
above  the  liquid  surface 
outside.  The  entire  sur- 
face tension  around  the 
interior  of  the  tube  where 

the    film    is    attached   is    

ZTrrT,  r  being  the  radius 
of  the  tube.    Let  6  be  the    rz^"_^ 
angle  of  contact  which  the 
film  makes  with  the  wall 
of  the    tube.     Then   the 

vertical  component  of  the  tension,  pulling  the  liquid  up 
and  the  tube  down,  is  2jrrT  cos  6.  This  force  is  in  equi- 
librium with  the  weight  of  the  liquid  column  of  height  A. 
Let  d  be  the  density  of  the  liquid.  Then  the  weight  of 
the  column  is  7rr*hdg.  Consequently 

2>7rrT  cos  6  =  Trrhdg. 
2Tcos  0 


Therefore 


rdg 


or  the  elevation  is  inversely  proportional  to  the  radius  or 
diameter  of  the  tube. 

If  the  angle  of  contact  is  more  than  90°,  cos  9  is  nega- 
tive, and  the  elevation  becomes  a  depression.  If  the 
liquid  wets  the  tube  there  is  an  elevation ;  otherwise  there 
is  a  depression.  The  curved  surface  of  the  liquid  in  the 
tube  is  called  the  meniscus. 

For  water  the  angle  of  contact  with  clean  glass  is  sup- 
posed to  be  nearly  or  quite  zero.  Hence  in  this  case 

h  =  ^. 

rdg 
For  two  plates  at  a  distance  u  from  each  other  the  total 


122 


MECHANICS. 


tension  for  unit  length  along  the  plates  is  2T7,  and  the 
vertical  component  is  2^008  6.  The  weight  of  the  liquid 
column  of  cross-section  u  is  uhdy.  Hence 


or 


j      2TcosB 

n  =  -  —  -  -  . 
udg 


The  elevation  is  therefore  half  as  great  as  for  a  tube 
whose  diameter  is  u. 


97.    The   Normal   Pressure  on   a  Curved  Film.  —  A 
stretched  film  with  a   curvature    must   always    exhibit  a 

normal     pressure     directed 
toward^he  concave  side. 

Let  ab  (Fig.  59)  be  a 
small  portion  of  the  section 
of  a  cylindrical  film.  T  and 
T  represent  the  surface  ten- 
sion, stretching  this  film  of 
unit  width  perpendicular  to 
the  plane  of  the  paper,  and 
they  are  directed  tangen- 

tially  at  a  and  b.  Complete  the  parallelogram  and  their 
resultant  is  the  diagonal  NM.  This  is  the  normal  press- 
ure. Call  it  P'.  Then 

P'  =  ZT  sin  J<£. 

But  sinJ0  =  ^-,  R  being  the  radius  of  curvature  of 

Zt£li 

the  film  or  the  radius  of  the  cylinder.     Therefore 


If  ab  is  unity,  then  the  surface  of  the  film  considered  is 
one  square  unit,  and  the  normal  pressure  per  unit  surface 

T 

is  P=  —  ,  or  T  times  the  curvature. 


MECHANICS    OF  FLUIDS.  128 

Any  other  curved  surface  may  always  have  its  curva- 
ture expressed  at  any  point  by  two  principal  radii  of  cur- 
vature, the  planes  of  these  curvatures  being  at  right  angles 
to  each  other.  Let  their  radii  be  R  and  Rr  Then  the 
normal  pressure  is  the  pressure  due  to  the  two  curvatures 
conjointly,  or 


If  the  film  is  plane,  then  both  R  and  R±  are  infinite.  If 
both  sides  of  the  film  are  free  and  the  film  is  still  curved, 
then  the  normal  pressure  is  necessarily  zero,  and 

1  +  1  =  0. 
R     El 

This  can  be  true  only  when  R  =  —  /A,  that  is,  the  radii 
are  numerically  equal  and  the  centres  of  curvature  are  on 
opposite  sides  of  the  film.  Such  a  film  is  saddle-shaped, 
and  it  may  easily  be  obtained  by  means  of  an  oblong  loop 
of  wire,  bent  so  that  it  does  not  lie  in  a  plane. 

T 

For  a  soap  bubble  P  —  4—,  since  there  are  two  concen- 

jLii  . 

trie    spherical    surfaces.      Hence    such   a   bubble    always 
shrinks  when     the    interior  _ 

communicates  with  the  outer 
air  on  account  of  the  com- 
pression normally.  The  air 


inside  a  closed  bubble  must  Fig  60 

then  be  denser  than  the  outer 

air,  and  minute  vesicles  of  water  filled  with  air  are  still 

heavier  than  the  air  displaced  by  them. 

The  normal  pressure  accounts  for  the  motion  of  drops 
of  liquid  in  conical  capillary  tubes.  Thus  a  drop  of  water 
introduced  into  the  larger  end  of  a  glass  tube  will  move 
toward  the  smaller  end  (Fig.  60),  while  a  globule  of 


124  MECHANICS. 

mercury  introduced  into  the  smaller  end  will  move  toward 
the  larger. 


98.    The  Angle  of  Contact  (A.  and  B.,  95  ;  B.,  206). - 

Suppose  the  three  dividing   surfaces    of   three  fluid  sub- 
stances in  contact  to  meet  along  the  line  through  0  (Fig. 

61),  perpendicular  to  the  plane 
of  the  paper.    Let  Tab  be  the  sur- 
face tension  between  the  media 
a  and  5,  Tac  that  between  media  a 
and  c,  and  Tbc  that  between  me- 
dia b  and  c.     Then  if  the  three 
tensions  are  in  equilibrium,  the 
angles   between  them    are  con- 
stant ;  for  the  three  tensions  may  be  represented  by  the 
three  sides  of  a  triangle  taken  in  order,  and  the  angles  be- 
tween the  three  surfaces  depend  only  upon  the  magnitude 
of  the  three  relative  surface  tensions. 

But  if  Tab  is  greater  than  the  sum  of  TM  and  Tbc,  then 
the  angle  between  Tac  and  Tlc  becomes  zero,  and  the  fluid 
c  spreads  itself  out  in  a  thin  sheet  between  a  and  b.  This 
is  the  case  with  oil  between  air  and  water.  Let  a  be  air, 
b  water,  and  c  oil.  Then 

Tab  =  81       dynes. 
Tac  =  36.88      " 
2^  =  20.56      " 

Hence  Tab>  Tac  +  Tbc, 

or  81  >  (20.56  +  36.88). 

Hence  a  drop  of  oil  on  the  surface  of  water  cannot  be 
in  equilibrium,  but  spreads  itself  out  indefinitely  thin  be- 
tween the  air  and  water. 


MECHANICS   OF  FLUIDS.  125 

When  two  fluids  a  and  b  (Fig.  62)  are  in  contact  with 
a  plane  solid  c,  and  their  surface  of  separa-     y>ac 
tion  makes  an  angle  6  with  the  solid,   the 
equation  of  equilibrium  is 

Tac  =  Tlc  +  T*  cos  0. 

But  if  Tac  be  greater  than  the  sum  of  Tbc 
and  Ta6,  the  equation  gives  an  impossible 
value  for  cos-  0,  the  angle  becomes  evanes- 
cent, and  the  fluid  b  spreads  out  and  wets 
the  surface  c.  A  drop  of  water  will  in  this 
way  spread  out  over  the  surface  of  a  clean  horizontal  plate 
of  glass ;  while  a  drop  of  mercury  will  gather  itself  to- 
gether till  the  edges  make  a  fixed  angle  with  the  plate. 

99.  Superficial  Viscosity  (D.,  258) Another  prop- 
erty of  liquid  films,  independent  of  surface  tension,  is  their 
superficial  viscosity.  Surface  tension  is  a  constant  stress 
in  the  bounding  surface  of  a  liquid,  while  superficial  vis- 
cosity is  a  sort  of  surface  friction  which  manifests  itself 
only  when  something  acts  to  rupture  or  otherwise  disturb 
the  surface  film.  A  solution  of  saponine  exhibits  super- 
ficial viscosity  to  a  marked  degree.  If  a  small  magnetic 
needle  be  floated  on  the  surface  of  it,  the  needle  will 
remain  in  any  position  because  the  earth's  magnetic  direc- 
tive force  is  unable  to  drag  the  liquid  film  around  with  the 
needle. 

In  most  other  liquids,  when  the  needle  turns,  it  carries 
with  it  the  whole  surface  film,  as  may  be  shown  by  strew- 
ing on  the  surface  lycopodium  powder. 

The  viscosity  of  the  surface  film  is,  as  a  rule,  much 
greater  than  the  viscosity  in  the  interior  of  the  liquid. 

Superficial  viscosity  holds  a  bubble  on  the  surface  of  a 
liquid  together,  while  the  contraction  of  the  surface  due 


126  MECHANICS. 

to  surface  tension  tends  to  break  it.  Soapy  water  makes 
good  bubbles,  because  while  its  surface  tension  is  small 
its  surface  viscosity  is  large.  A  bubble  rising  through 
the  liquid  will  raise  a  film  at  the  surface  which  the  sur- 
face tension  cannot  break. 

Pure  water  has  large  surface  tension,  and  relatively 
small  superficial  viscosity.  Hence  it  does  not  froth. 

Oil  has  small  surface  tension,  but  large  surface  viscosity 
or  tenacity. 

To  this  fact  must  be  attributed  the  stilling  of  the  sea 
when  oil  is  poured  on  it.  The  new  surface  is  relatively 
tenacious,  and  it  is  not  readily  broken  into  surf  by  the 
pressure  of  the  waves  from  beneath. 

"  The  superficial  film  of  a  liquid  is  thus  seen  to  be  a  seat 
of  energy,  and  to  be  physically  different  from  the  interior." 

100,  Air  has  Weight.  —  Aristotle  attempted  to  deter- 
mine whether  air  had  weight  by  weighing  a  bladder  col- 
lapsed and  then  inflated.    Of  course  the  change  of  buoyancy 
in  the  two  cases  offset  the  difference  due  to  the  weight  of 
the  air  removed. 

Galileo  determined  that  water  would  not  rise  above  32 
feet  in  the  pumps  of  the  Duke  of  Tuscany. 

Since  the  invention  of  the  air-pump  it  has  been  deter- 
mined that  air  and  hydrogen  have  the  following  weights : 
1  litre  of  air  weighs  1.2759  gms., 
1  litre  of  hydrogen  weighs  0.08837  gms., 
both  at  a  temperature  of  0°  C.  and  under  a  pressure  of 
106  dynes. 

101,  Atmospheric   Pressure.  —  The  total  pressure  of 
the  atmosphere  was  first  determined  by  Torricelli  in  1643. 

He  took  a  glass  tube,  a  little  less  than  a  metre  long  and 


MECHANICS    OF  FLUIDS. 


127 


closed  at  one  end,  and  filled  it  with  mercury.  Closing  the 
upper  end  with  the  thumb  he  inverted  the  tube  and  placed 
the  lower  end  under  mercury  contained  in  another  vessel 
(Fig.  63).  On  removing  the  thumb  the  mercury  fell  in 
the  tube,  and  came  to 
rest  at  a  height  of  about 
76  cms.  above  the  mer- 
cury surface  in  the 
outer  vessel,  leaving 
a  vacuum  in  the  tube 
above  it,  which  has  since 
been  known  as  a  Torri- 
cellian vacuum.  It  was 
rightly  concluded  by 
Torricelli  that  the  mer- 
cury is  sustained  in  the 
tube  by  the  pressure  of 
the  atmosphere  on  the 
mercury  surface  exter- 
nal to  the  tube. 

Pascal  performed  two 
experiments  to  demon- 
strate that  the  column 
of  mercury  is  supported 
by  atmospheric  pressure. 
In  the  first  the  mercury  was  replaced  by  lighter  liquids, 
with  the  result  that  the  height  of  the  sustained  column 
was  always  inversely  proportional  to  the  density  of  the 
liquid. 

In  the  second  experiment  Pascal  had  the  mercury 
column  carried  to  the  top  of  the  Puy-de-Dchne,  about 
1,000  metres  high.  The  pressure  of  the  atmosphere  being 
less  on  top^of  the  mountain,  it  was  anticipated  that  the 


Fig.  63. 


1^8  MECHANICS. 

mercury  column  would  fall.     A  fall  of  nearly  eight  centi- 
metres was  observed. 

The  apparatus  of  Torricelli,  when  provided  with  a  scale 
for  the  purpose  of  reading  the  height  of  the  column  of 
mercury,  is  called  a  barometer. 

Atmospheric  pressure  on  a  square  centimetre  of  surface 
is   therefore   the   weight  of    the   column  of  mercury  one 
square  centimetre  in  cross-section,  and  76  centimetres  in 
height  at  a  temperature  of  0°  C.     This  is 
76x13.596  =  1033,3  gms., 
or  1033.3  x  980  =  1,012,630  dynes. 

This  is  a  little  more  than  106  dynes,  a  megadyne. 

1O2.  Height  of  the  Homogeneous  Atmosphere.  --If 
the  atmosphere  were  of  the  same  density  throughout  its 
entire  height  as  at  the  earth's  surface  this  height  in  centi- 
metres could  be  determined  in  terms  of  pressure,  density, 
and  gravity.  It  is  often  called  the  height  of  the  homo- 
geneous atmosphere.  The  pressure  on  one  square  centi- 
metre would  be 

P  =  Hdg. 

Whence  H=  —  . 

dg 

P  is  the  pressure  of  the  atmosphere  in  dynes  per  square 
centimetre  and  d  is  the  density  of  the  air  at  0°  C.  and  a 
pressure  of  76  cms.  of  mercery.  Hence 


For  the  same  temperature  at  different  elevations  P 
varies  directly  as  d.  H  therefore  remains  of  the  same 
value  except  for  the  change  in  g. 

103.  Boyle's  Law  (B.,  190;  A.  and  B.,  141).  —  The 
law  governing  the  compressibility  of  gases,  at  a  constant 


MECHANICS    OF  FLUIDS.  129 

temperature,  was  discovered  by  Robert  Boyle  in  1660.  It 
is  commonly  known  as  Boyle's  law,  though  it  is  sometimes 
ascribed  to  Mariotte.  The  law  is  as  follows  :  The  volume 
of  a  given  mass  of  gas,  at  a  constant  temperature,  is  in- 
versely as  the  pressure  to  which  it  is  subjected.  Expressed 
in  symbols  it  is 

V  T)1 

__  =  JL    or  pv  —  »V. 

VI  p 

According  to  the  law  the  product  pv,  at  one  tempera- 
ture, is  a  constant. 

Since  volumes  are  inversely  as  densities,  or 

L-^' 

vf      d* 

it  follows  that  ^-  =  £-  , 

d/     p' 

or  the  densities  are  directly  proportional  to  the  pressures 
to  which  the  gas  is  subjected. 

Boyle's  law  is  only  approximately  true.  Such  gases  as 
sulphur  dioxide,  chlorine,  and  carbon  dioxide,  which  are 
most  easily  liquefied  by  pressure,  depart  from  the  law  most 
widely.  Near  the  point  of  liquefaction  the  product  pv  is 
much  smaller  than  Boyle's  law  requires. 

Such  gases  as  hydrogen,  oxygen,  and  nitrogen  follow  the 
law  most  closely.  These  gases  cannot  be  liquefied  except 
by  the  combined  means  of  great  reduction  of  temperature 
and  great  pressure.  They  cannot  be  reduced  to  a  liquid 
at  ordinary  temperatures  by  any  pressure,  however  great. 

They  show  a  departure  from  Boyle's  law  different  from 
that  of  the  other  class  of  gases.  For  every  gas  of  this 
class,  except  hydrogen,  a  minimum  value  of  pv  has  been 
found,  beyond  which,  if  the  pressure  is  increased,  the 
product  pv  increases.  Thus  if  the  product  pv  is  taken 
as  unity  for  air  under  a  pressure  of  one  atmosphere,  at 


130 


MECHANICS. 


77  atmospheres  it  becomes  0.9803;  at  176  atmospheres, 
1.0113;  at  400  atmospheres,  1.1897.  The  minimum  value 
ofjtwisat  77  atmospheres.  If  the  experiments  are  made 
at  a  higher  temperature  the  pressure  at  which  the  mini- 
mum value  of  pv  occurs  is  greater  and  the  agreement  with 
the  law  is  closer. 

No  minimum  value  of  pv  for  hydrogen  has  been  found, 
and  this  value  must  occur,  if  at  all,  for  pressures  less  than 
one  atmosphere.  The  value  of  pv  for  hydrogen  is  always 
greater  than  Boyle's  law  requires. 

V 

104.  The  Air-Pump.  —  The  air-pump  was  invented  by 
Otto  von  Guericke.  Its  action  depends  upon  the  elastic 
force  of  the  gas  by  which  it  tends  to  expand  indefinitely. 
Fig.  64  shows  the  essential  parts  of  one  of  the  best  forms 
of  the  present.  A  piston  P,  with  a  valve  S  in  it,  works 


Fig-  64. 


in  a  cylinder  communicating  with  the  air  by  a  valve  at 
its  upper  end  opening  outward,  and  with  the  receiver  E 
by  a  valve  S'  at  its  lower  end.  When  the  piston  ascends 
8  is  closed  and  &  is  open.  The  gas  expands  and  fills  the 
cylinder.  During  the  downward  stroke  S  is  open  and  & 


MECHANICS    OF  FLUIDS.  131 

is  closed.  The  gas  thus  escapes  above  the  piston,  and  is 
forced  into  the  open  air  when  it  is  sufficiently  compressed 
on  the  upward  stroke  to  open  the  outward-opening  valve. 
Let  v  be  the  volume  of  the  receiver  E,  and  c  that  of  the 
pump  cylinder.  Let  d  and  di  be  the  densities  of  the  gas 
in  the  receiver  before  and  after  the  first  stroke  and  dn  the 
density  of  the  gas  remaining  after  the  wth  stroke.  Then 
since  the  volume  v  becomes  v  +  c  at  the  end  of  the  first 
stroke,  by  Boyle's  law 


d         v  +  c 
For  the  second  stroke 


dl       v  +  e 
Therefore  after  two  strokes 


again, 


3" 

After  n  strokes 


n 


This  expression  can  become  zero  only  after  an  infinite 
number  of  strokes.  But  the  limit  of  exhaustion  by  the 
mechanical  air-pump  is  reached  after  a  moderate  number 
of  strokes  for  several  reasons.  Among  them  are  leaks  at 
the  valves  and  around  the  piston,  untraversed  space  above 
and  below  the  piston,  and  air  absorbed  by  the  oil  used  for 

lubrication. 
r-       .^ 

105.  Correction  to  the  Weight  of  Bodies  for  Buoyancy 
of  Air.  —  The  apparent  weight  of  a  body  is  less  than  its 
real  weight  by  the  weight  of  the  air  which  it  displaces. 
But  since  this  applies  equally  well  both  to  the  weights  and 


132  MECHANICS. 

to  the  body  weighed,  a  correction  must  be  found  depend- 
ing upon  their  relative  densities. 

Let  x  be  the  real  mass  of  the  body  in  grammes. 

Let  w  be  the  real  mass  of  the  weights. 

Let  d  be  the  density  of  the  body. 

Let  B  be  the  density  of  the  weights. 

Let  a  be  the  density  of  the  air. 

Then  the  volume  of  the  body  is  ^  . 

ct 

The  volume  of  the  weights  to  counterbalance  is^. 

o 

The  masses  of  air  displaced  by  the  two  are   therefore 

-a  and  -a  respectively. 
a  o 

For  equilibrium  the  equation  is 
a  a 

X  --  X  =  W  —  -IV. 

d  o 

a 

^  (  /1      ~l\  ) 

Whence  x  —  w  —     -  =  w  \  1  +  a  (  —  -  )  [  nearly. 
\—°L          '  \d     B/  } 

d 

If  d  is  greater  than  8  the  correction  is  negative,  that 
is,  the  real  weight  is  less  than  the  apparent  weight.  If  d  is 
less  than  S,  the  correction  is  positive.  The  correction  is 
zero  only  when  d  and  S  are  equal  to  each  other,  or  when  the 
body  weighed  and  the  weights  are  of  the  same  density. 

For  example,  let  a  mass  of  sulphur,  density  2,  have  an 
apparent  weight  of  100  gms.  when  weighed  with  brass 
weights  of  density  8.4.  The  correction  is  then  positive, 
and  the  true  weight  is 

*  =  WO     1+-1        =  100440  go*. 


The  correction  is  0.049  gin. 


MECHANICS   OF  FLUIDS.  133 

• 
106.    Torricelli's  Theorem  for  the  Velocity  of  Efflux. 

•  —  Let  a  small  opening  be  made  in  the  side  of  a  vessel 
containing  water,  the  depth  of  the  orifice  below  the 
surface  being  h.  Torricelli's  formula  for  the  velocity  of 
efflux  is 

v2  =  Zgh. 

This  is  the  velocity  which  a  heavy  body  would  acquire 
in  falling  through  the  height  h  in  a  vacuum. 

If  we  suppose  a  small  mass  m  to  issue  from  the  orifice, 
an  equal  mass  must  have  fallen  some  distance  a\  to  take 
its  place.  Then  another  equal  mass  must  have  fallen  a 
distance  «2,  and  so  on  through  a  series  to  the  surface. 

The  total  loss  of  potential  energy  is 


where  h  is  the  sum  of  ai  ,  a2  ,  as  ,  etc. 

If  the  loss  in  potential  energy  is  all  represented  by  the 
energy  of  motion  acquired  by  the  mass  m,  we  may  write 
%mv2  —  mgh. 

From  this  equation  we  obtain 


which  is  Torricelli's  formula. 

If  a  square  centimetres  is  the  area  of  the  orifice,  the 
number  of  cubic  centimetres  of  water  flowing  out  in  t 
seconds  should  be  avt,  if  Torricelli's  formula  is  right.  The 
quantity  is  always  smaller  than  this  because  the  effective 
area  of  the  stream  is  not  the  area  of  the  orifice,  but  the 
cross-section  of  the  smallest  part  of  the  stream,  or  the 
vena  contracta,  which  is  about  62  per  cent  of  the  area  of 
the  orifice.  The  conical  shape  of  the  issuing  jet  is  due  to 
the  lateral  pressure  on  the  water  as  it  approaches  the 
orifice.  If  a  short  cylindrical  tube,  whose  length  is  two 
or  three  times  its  diameter,  be  placed  in  the  opening  so  as 


134  MECHANICS. 

to  project  outwards,  the  flow  is  increased  to  about  82  per 
cent  of  the  theoretical  amount. 

107.    Range  of  Jets.  —  Let  ED  (Fig.  65)  be  the  side  of 
a  vessel  of  water,  and  let  the  surface  be  at  E.     Also  let 

all    the    distances   EA, 

%  AB,    BO,   and    CD  be 

equal  to  h.  Let  v  be 
the  velocity  of  efflux 
from  the  o  r  i  fi  c  e  A. 
Then  the  range  a  of  this 
stream  will  be 
a  =  vt, 

in  which  t  is  the  time  of 
falling  to  the  horizontal  plane  through  D.     Also 

b  = 


But  if  v2  =  2gh,  a2  =  2c/ht\  and  t2=  __. 

2ffh 

Substitute  this  value  of  t  in  the  equation  for  b  and 

*       a* 
=  5' 

or  4hb  —  a2  and  a  = 


It  follows  that  if  h  and  b  exchange  values,  the  range 
will  be  the  same,  for  their  product  bh  is  unaltered.  This 
means  that  the  range  a  of  the  jet  from  O  is  the  same  as  of 
the  one  from  A. 

When  the  sum  of  two  factors  is  a  constant,  their  product 
is  a  maximum  when  the  factors  are  equal  to  each  other  ; 
the  range  a  is  therefore  greatest  for  the  opening  B  at.  the 
middle  of  the  height,  for  then  h  and  b  are  equal  to  each 
other,  and  their  sum  is  the  constant  ED. 


MECHANICS    OF  FLUIDS. 


135 


This  may  be  shown  in  another  way.  On  ED  as  a 
diameter  describe  a  semicircle.  Then  the  square  of  any 
half-chord  equals  the  product  of  the  two  sections  of  the 
diameter;  therefore 

c?  =  bh  or  c  — 


Hence  \fbli  will  be  a  maximum  when  c  is  a  maximum 
or  at  the  point  B.  But  the  range  a  is  2V7>A,  and  is  there- 
fore greatest  for  the  opening  at  B. 

The  form  of  the  parabolic  streams  shows  whether  v  has 
a  value  corresponding  to  Torricelli's  formula.  It  is  found 
to  differ  not  more  than  one  per  cent  from  Torricelli's  value 


1O8.  The  Common  Siphon.  —  Let  y  (Fig.  66)  be  the 
height  of  the  highest  point  of  the  siphon  above  the  surface 
of  the  liquid  in  the  vessel  from  which 
the  discharge  takes  place  ;  and  let  x  be 
the  height  of  the  same  point  of  the 
siphon  above  its  open  end  or  the  lo"wer 
surface  of  the  liquid  if  the  longer  arm 
dips  below  the  liquid.  Let  H  be  the 
height  of  a  column  of  the  liquid  equal 
to  atmospheric  pressure.  Then  the 
pressure  011  one  square  centimetre  of 
the  highest  cross-section  of  the  siphon 
outwards  is  d(H—y)-,  and  the  pres- 
sure inwards  on  the  same  area  is 
d^H—x).  The  difference  is  the  effective  pressure,  or 
head,  producing  the  flow, 

—}  —  dH—x=dx  —       =dh. 


Fig.  66. 


The  density  of  the  liquid  is  represented  by  d.     In  the 
case  of  water  d  is  unity  and  the  head  is  h. 


136  MECHANICS. 

If  y  exceeds  H,  the  liquid  will  not  rise  to  the  bend  of 
the  siphon  by  atmospheric  pressure,  and  the  flow  ceases. 

1O9.  Mariotte's  Flask.  —  Marietta's  flask  (Fig.  67)  is 
an  arrangement  to  secure  uniform  velocity  of  efflux.  The 
flask  has  a  tubulure  near  the  bottom  as  an 
outlet.  A  glass  tube  passes  through  an  air- 
tight  stopper  and  extends  down  to  within 
a  distance  h  of  the  horizontal  plane  through 
the  outlet  o.  The  flow  will  then  continue 
with  a  head  h  until  the  water  descends  to 
the  lower  end  of  the  tube. 

For  the  pressure  at  the  lower  end  of  the 
tube  is  the  pressure  of  one  atmosphere,  the 
-o        same  as  at  the  opening  o.     As  water  flows 


Fjg  67  out  air   enters  by  the  tube  and  takes    its 

place,  so  that  the  effective  pressure  remains 
h.  The  pressure  of  the  air  in  the  flask  and  of  the  column 
of  water  above  the  lower  end  of  the  tube  together  con- 
stantly equal  the  pressure  of  the  atmosphere. 

PROBLEMS. 

1.  A  uniform  circular  cylinder  weighing  50  kilos,  has  a  radius  of 
25  cms.  and  revolves  without  friction  around  a  horizontal  axis.     A 
thread  rolled  around  the  cylinder  carries  at  the  free  end  a  weight  of 
one-half  a  kilogramme.     Find  how  far  the  weight  will  descend  from 
rest  in  three  seconds. 

2.  A  piece  of  silver  and  a  piece  of  gold  are  suspended  from  the 
two  ends  of  a  balance  beam  with  equal  arms.     The  beam  is  in  equi- 
librium when  the  silver  is  immersed  in  alcohol  (sp.gr.  0.85),  and 
the  gold  in  nitric  acid  (sp.  gr.  1.5)  ;  if  the  densities  of  the  gold  and 
silver  are  19.3  and  10.5  respectively,  what  are  their  relative  masses? 

3.  The  length  of  the  seconds  pendulum  at  the  equator  being 
990.93  inms.,  what  is  the  corresponding  value  of  g  ?     What  would 
be  the  value  of  (j  there  if  the  earth  did  not  rotate  ? 


NATURE    AND    MOTION    OF    SOUND.  137 


SOUND. 


CHAPTER    VI. 

NATURE    AND   MOTION    OF   SOUND. 

HO.  Sound  and  Hearing.  —  In  all  perceptions  by  the 
senses  it  is  necessary  to  distinguish  between  the  sensations 
themselves  and  the  external  cause  of  them  —  between  the 
subjective  and  the  objective  aspects  of  the  phenomenon. 
It  is  important  to  remember  also  that  the  objective  causes 
of  our  sensations  bear  no  resemblance  to  the  sensations 
themselves.  The  external  stimulus  stands  first  in  the 
series  of  energy-changes  leading  to  a  sensation,  but  it 
is  not  like  the  sensation.  Sound  should  therefore  be  dis- 
tinguished from  hearing  in  the  study  of  sound  as  a  branch 
of  Physics.  All  the  external  phenomena  of  sound  may  be 
present  without  the  hearing  ear. 

All  questions  concerning  sound  come  ultimately  for 
decision  to  the  organ  of  hearing ;  but  in  referring  our  sen- 
sations of  sound  to  their  external  cause,  we  are  only 
interpreting  signs  presented  to  consciousness,  and  drawing 
conclusions  from  them  respecting  outward  phenomena. 
When  this  process,  controlled  by  observation,  experience, 
and  trained  reasoning,  has  led  to  the  discovery  of  the 
physical  facts  constituting  the  foundation  of  sound,  our 
investigations  are  largely  transferred  to  the  domain  of 
mechanics  (Lord  Rayleigli). 


138  SOUND. 

111.  The  Source  of  Sound  a  Vibrating  Body.  —  Very 
cursory  examination  serves  to  show  that  the  source  from 
which  sound  proceeds  is  always  a  vibrating  body.  "  Sound 
and  movement  are  so  correlated  that  one  is  strong  when 
the  other  is  strong,  one  diminishes  when  the  other  dimin- 
ishes, and  the  one  stops  when  the  other  stops."  1 

Any  regular  succession  of  taps  produces  a  musical 
sound.  The  element  of  regularity  or  periodicity  is  essen- 
tial to  make  it  musical.  Otherwise  it  is  mere  noise.  When 
a  heavy  toothed  wheel  is  rotated  and  a  card  is  held  against 
the  teeth  a  musical  sound  of  definite  pitch  is  produced. 
So  also  the  sound  produced  by  a  circular  saw  is  musical  at 
a  distance  where  the  highly  discordant,  irregular  elements 
are  eliminated. 

If  a  goblet,  partly  filled  with  water,  is  set  vibrating  by 
drawing  a  bow  across  its  edge,  the  tremors  of  the  glass  are 
communicated  to  the  water  and  throw  its  surface  into 
violent  agitation  in  four  sectors,  with  intermediate  areas 
of  relative  repose.  All  this  ceases  with  the  subsidence  of 
the  sound. 

If  a  mounted  tuning-fork  is  sounded  and  a  light  ball  of 
pith  or  ivory,  suspended  by  a  thread,  is  brought  in  contact 
with  one  of  the  prongs  at  the  back,  it  will  be  violently 
thrown  away  by  the  energetic  vibrations. 

If  a  minute  globule  of  mercury  is  attached  to  a  stretched 
wire  by  means  of '  a  little  grease  and  lampblack,  and  is 
examined  with  a  microscope,  it  will  be  found  in  motion, 
describing  a  line  backward  and  forward,  so  long  as  the 
wire  produces  a  musical  sound. 

A  stout  glass  tube,  several  feet  in  length,  may  be  made 
to  emit  a  musical  sound  by  grasping  it  by  the  middle 
and  briskly  rubbing  one  end  with  a  moistened  cloth.  So 

1  Blaserna  on  Sovnd,  p.  7. 


NATURE    AND    MOTION    OF    SOUND.  139 

energetic  are  the  longitudinal  vibrations  excited  that  it  is 
not  difficult  to  break  the  tube  near  the  hand,  and  on  the 
side  opposite  to  the  end  rubbed,  into  many  very  narrow 
rings. 

The  vibrating  body  producing  sound  may  be  solid, 
liquid,  or  gaseous.  Only  the  first  and  last  are  used  in 
musical  instruments,  the  first  comprising  all  instruments 
employing  strings,  reeds,  or  bars,  and  the  last  including 
wind  instruments  of  various  sorts. 

112.  The  Medium  of  Propagation.  —  Sound  requires 
for  transmission  to  the  ear  a  continuous,  ponderable,  elastic 
medium ;  for  the  vibrations  of  a  sonorous  body  cannot 
affect  the  organ  of  hearing  without  a  medium  of  com- 
munication between  them.  If  the  vibrating  body  be  isolated 
so  that  the  required  elastic  medium  does  not  extend  to 
the  source  of  vibrations,  no  sound  will  be  perceived.  This 
is  somewhat  imperfectly  demonstrated,  after  the  manner 
of  Otto  von  Guericke,  by  suspending  a  bell  by  a  thread  in 
a  receiver  from  which  the  air  can  be  exhausted.  The  bell 
must  not  be  allowed  to  communicate  its  vibrations  directly 
to  the  pump  plate.  The  sound  becomes  feebler  as  the  ex- 
haustion proceeds.  Finally,  if  hydrogen  be  admitted  and 
again  exhausted,  the  sound  will  cease  altogether,  though 
the  hammer  may  still  be  seen  to  strike  the  bell.  When 
the  medium  about  the  bell  is  entirely  removed  it  can  no 
longer  give  up  its  energy  to  surrounding  bodies,  so  as  to 
set  their  molecules  swinging  with  a  to-and-fro  motion ;  but 
its  vibratory  energy  is  converted  into  heat,  in  which  form 
it  can  be  propagated  outwards  by  means  of  the  ether.  It 
vibrates  longer  and  loses  its  energy  more  slowly  when  in 
an  exhausted  receiver  than  when,  it  is  beating  the  air. 
So  a  thin  platinum  wire  or  carbon  filament  continues  to 


140  SOUND. 

glow  for  several  seconds  after  the  electric  current  which 
heats  it  ceases  to  flow.  But  when  surrounded  by  an  at- 
mosphere it  cools  very  quickly,  because  it  gives  up  its 
heat-energy  to  the  surrounding  gas,  producing  convection 
currents.  The  rarer  the  air  at  the  source  the  feebler  will 
be  the  sound. 

The  transmission  of  sound  requires  a  medium  both 
elastic  and  ponderable  —  elastic,  because  elasticity  is  the 
property  by  means  of  which  the  motion  constituting  sound 
is  handed  on  from  particle  to  particle ;  and  ponderable, 
because  sound  is  not  transmitted  through  space  exhausted 
of  ordinary  gross  matter.  The  ethereal  medium  is  not 
concerned  in  the  transmission  of  sound. 

113.  Definition  of  Sound.  —  Sound  maybe  denned  as 
a  vibratory "  movement  excited   in   an    elastic  body,  and 
transmitted  to  the  ear  by  means  of  a  continuous,  elastic, 
ponderable  medium.     "Acoustics  has   for  its   object  the 
study  of    those  phenomena  which  may  be  perceived  by 
the  ear." 

114.  The  Transmission  of  Sound  (A.  and  B.,  354).  - 
The  oscillatory  motions  of  a  sounding  body  are  communi- 
cated to  the  air  as  the  usual  medium  of  transmission.    The 
vibrations  are  said  to  be  longitudinal ;  that  is,  in  the  direc- 
tion of  the  sound-transmission.     They  are  distinguished  in 
this  way  from   the  transverse  vibrations  of   water-waves, 
in  which  the  motions  of    the  particles  are  more  or  less 
nearly  at  right  angles  to  the  direction  in  which  the  waves 
are  running.     The  vibrations  of  light  are  transverse. 

The  manner  in  which  the  motion  is  transmitted  from 
particle  to  particle  may  be  illustrated  by  means  of  a  row 
of  elastic  balls  lying  in  contact  on  two  curved  rails  with 


NATURE    AND    MOTION    OF    SOUND.  141 

the  ends  elevated.  If  one  ball  is  allowed  to  roll  down  the 
groove  and  strike  against  the  first  one  in  line,  the  motion 
or  impulse  is  handed  on  through  the  whole  series,  and  the 
last  ball  moves  up  the  incline.  The  elasticity  of  the  balls 
explains  the  transfer  of  the  motion  through  the  series ; 
the  energy  of  the  motion  is  independent  of  the  elasticity 
of  the  conducting  medium.  It  must  all  be  supplied  at  the 
origin  of  the  motion. 

When  the  first  ball  strikes  the  second,  compression  takes 
place. .  The  elasticity,  called  into  activity  by  the  distortion 
of  the  balls,  tends  to  restore  them  to  their  unstrained  form. 
The  stress  of  elastic  recovery  is  the  same  in  both  direc- 
tions. The  backward  thrust  brings  the  first  ball  to  rest, 
while  the  forward  one  drives  the  second  ball  on  against 
the  third.  The  same  operation  is  repeated  between  the 
second  and  third  balls,  and  so  on  to  the  end  of  the*  series. 
But  the  last  ball,  not  having  any  to  which  it  can  give  up 
its  motion,  moves  off  up  the  incline. 

To  describe  the  motion  by  which  sound  is  transmitted 
let  AB  (Fig.  68)  represent  an  elastic  cylinder,  and  let  the 
layer  a  suffer  a  small  displacement  to  the  right.  The 


A6 


Fig.  68. 

effect  of  this  displacement  is  that  a  approaches  6,  produc- 
ing a  condensation  or  crowding  together  of  the  particles. 
Therefore  b  is  thrust  forward,  and  the  motion  is  commu- 
nicated in  the  same  manner  from  layer  to  layer  through 
the  cylinder. 

If  now  a  executes  regular  vibrations,  its  motions  will 

t 


142  SOUND. 

ultimately  be  communicated  to  all  the  other  layers, 
because  they  are  all  tethered  together  as  an  elastic 
medium,  and  in  time  each  layer  of  particles  will  be  execut- 
ing vibrations  similar  to  those  of  a. 

If  the  period  of  vibration  of  a  is  £,  and  the  speed  of 
transmission  is  v,  then  in  one  complete  vibration  of  a  the 
disturbance  will  travel  a  distance  *  =  vt,  or  to  of  in  the  figure. 
During  two  complete  vibrations  it  will  travel  a  distance 
2s  or  to  a"  ;  in  three  periods  to  a'",  and  so  on.  The  layer 
at  a'  therefore  begins  its  first  excursion  as  a  begins  its 
second  ;  a"  begins  its  first  as  a'  begins  its  second,  and  a  its 
third,  and  so  on.  The  layer  at  d,  midway  between  a  and 
a',  begins  its  first  vibration  as  a  completes  its  first  half- 
vibration  ;  and  it  therefore  moves  forward  while  a  moves 
backward.  This  related  movement  of  particles  in  the 
cylinder  constitutes  a  wave.  While  a  is  moving  forward 
the  particles  near  it  constitute  a  compression  ;  while  it  is 
moving  backward  they  constitute  a  rarefaction.  The  dis- 
tance aa',  a'a",  etc.,  traversed  by  the  disturbance  during 
the  period  of  a  complete  vibration  of  any  one  of  the  par- 
ticles, is  called  a  wave-length. 

115.  The  Motion  of  the  Particles  and  of  the  "Wave. 
—  The  motion  of  the  individual  particles  of  the  medium 
conveying  sound  is  quite  distinct  from  the  motion  of  the 
sound-wave  itself.  This  distinction  is  characteristic  of  all 
undulations  transmitted  through  a  medium  of  motion.  A 
sound-wave  is  composed  of  a  condensation  followed  by  a 
rarefaction.  In  the  former  the  particles  of  the  medium 
have  a  forward  motion  in  the  direction  in  which  sound  is 
travelling;  in  the  latter  they  have  a  backward  motion, 
while,  at  the  same  time,  both  condensation  and  rarefaction 
are  travelling  steadily  forward  with  a  speed  independent 


NATURE    AND    MOTION    OF   SOUND.  143 

of  that  of  the  air  particles.  The  independence  of  the  two 
motions  is  aptly  illustrated  by  a  field  of  grain  across 
which  waves,  excited  by  the  wind,  are  coursing.  No  con- 
fusion between  the  two  motions  is  here  possible,  because  each 
stalk  of  grain  is  securely  anchored  to  the  ground,  while 
the  wave  sweeps  onward.  Each  head  of  grain  in  front  of 
the  crest  of  the  wave  is  found  to  be  rising,  while  all  those 
behind  the  crest  are  at  the  same  time  falling.  They 
all  sway  fonvard  and  backward,  not  simultaneously,  but 
in  succession,  while  the  wave  itself  travels  continuously 
forward. 

In  a  sound-wave,  therefore,  the  motion  of  the  wave 
and  the  motion  of  the  particles  composing  the  wave  are 
not  identical ;  a  wave  in  air  is  in  no  sense  a  current ;  the 
motion  of  the  condensation  and  of  the  particles  composing 
it  are  in  the  same  direction ;  while  the  motion  of  the  rare- 
faction and  of  the  particles  in  it  are  in  opposite  directions. 
The  transmission  of  a  wave  is  the  transmission  of  energy, 
and  not  the  transfer  of  the  medium  composing  the  wave. 
A  series  of  particles  along  a  line  marking  the  progress  of 
the  wave  are  in  successively  different  phases  of  their 
motion ;  and  the  distance  between  two  particles  having 
the  same  phase  is  a  wave-length. 

While  any  element  of  the  medium  merely  oscillates 
about  its  position  of  rest,  there  is  a  continuous  handing  on 
or  flow  of  the  energy  from  point  to  point.  In  the  case  of 
a  current,  matter  flows  from  one  place  to  another,  carrying 
the  associated  energy  with  it,  so  that  there  is  a  flow  of 
both  energy  and  matter. 

A  wave-front  is  the  continuous  locus  of  all  points  which 
are  in  the  same  phase  of  vibration,  or  of  those  portions  of 
the  medium  which  at  the  instant  considered  are  equally 
and  similarly  distorted. 


144  SOUND. 

116.  Experimental  Determination  of  the  Velocity  of 
Sound  in  Air.  —  Since  the  motion  of  the  particles  of  the 
medium  is  distinct  from  the  motion  of  the  sound-wave, 
the  two  kinds  of  motion  admit  of  independent  treatment 
and  illustration.  We  shall  consider  first  the  velocity  of 
sound  in  air  and  other  media. 

The  usual  determination  of  the  velocity  of  sound  is 
founded  upon  the  measurement  of  the  interval  which 
elapses  between  the  observation  of  some  phenomenon  first 
by  sight  and  then  by  hearing.  The  observations  have  com- 
monly been  those  of  the  flash  and  the  report  of  a  distant 
cannon.  Since  light  is  transmitted  with  such  rapidity,  the 
interval  between  the  two  observations  may  be  regarded 
without  sensible  error  as  that  which  the  sound  actually 
requires  to  traverse  the  distance  between  the  two  stations. 
The  earlier  observations  in  the  latter  part  of  the  seven- 
teenth century  and  at  the  beginning  of  the  eighteenth 
were  not  of  sufficient  value  to  report  here.  But  beginning 
with  those  made  by  the  French  Academy  of  Sciences  the 
following  are  some  of  the  most  trustworthy  results  : 

1.  Academy  of  Sciences,  1738 .  332.00  metres. 

2.  Bureau  des  Longitudes,  1822 331.00 

3.  Moll  and  Van  Beck,  1823 332.25       " 

4.  Stampfer  and  Myrbach,  1823 332.44       " 

5.  Bravais  and  Martins,  1844 332.37       " 

6.  Stone,  1871 332.40 

The  more  precise  measurements  give  a  velocity  of  332.4 
metres  at  a  temperature  of  0°  C.  All  of  the  above  results 
have  been  reduced  to  this  temperature. 

In  Stone's  determination  a  cannon  was  fired,  and  two 
observers,  three  miles  apart,  gave  signals  by  electricity 
on  hearing  the  report.  The  eye  observations  were  thus 


NATURE    AND    MOTION    OF    SOUND.  145 

eliminated,  as  well  as  the  perturbing  effect  of  the  violent 
disturbance  near  the  source  of  the  sound.  Still  two 
observers  were  necessary,  and  the  reflex  time  in  the  two, 
required  to  perceive  and  to  record  the  observation,  may 
not  have  been  the  same. 

From  1862  to  1866  Regnault  made  a  long  series  of  obser- 
vations on  the  transmission  of  a  shock  or  pulse  through 
the  water-pipes  of  Paris.  The  velocity  was  found  to  be 
somewhat  less  than  in  the  open  air.  The  disturbances  at 
each  station  were  recorded  automatically  by  electricity. 
The  principal  conclusions  may  be  summarized  as  follows : 1 

1.  In  a  cylindrical  pipe  the  intensity  of  the  wave  does 
not  remain  constant,  but  is  enfeebled  with  the  distance, 
and  the  more  rapidly  the  smaller  the  pipe. 

2.  The  velocity  of  sound  diminishes  at  the  same  time 
as  the  intensity.     In  a  conduit  1.1  metres  in  diameter  the 
velocity  was  334.16  for  a  distance  of   749.1  metres,  and 
330.52  for  a  distance  of  19,851.3  metres. 

3.  The  velocity  tends  toward  a  limit,  which  is  larger 
the  larger  the  pipe.     This  fact  is  exhibited  in  the  follow- 
ing table  : 

Diameter  of  conduit.  Velocity  at  zero.  Distance  traversed. 

0.108  m 326.66 4055.9 

0.216  " 328.18 6238.9 

0.300  " 328.96  ....  15240.0 

1.100  " 330.52  .....  19851.3 

After  all  corrections  had  been  made  Regnault  obtained 
for  the  limiting  velocity  330.6  metres  at  0°  C. 

4.  The    mode    of   production    of    the    wave  does    not 
appear  to  have  any  sensible  effect  on  the  speed  of  propa- 
gation. 

1  Violle's  Cours  de  Physique,  Tome  II.,  67. 


146  SOUND. 

5.     The  speed  of  propagation  in  a  gas  is  the  same,  what- 
ever may  be  the  pressure  to  which  the  gas  is  subjected. 

117.    Theoretical   Determination    of  the   Velocity   of 

Sound  (Phil.  Trans.,  187O,  277;  Maxwell's  Heat,  223). 

-  Let  A!  A2  (Fig.  69)  be  a  tube  of  one  square  centimetre 


Fig.  69. 

cross-sectional  area  and  of  indefinite  length.  Let  Al  and 
A2  be  two  imaginary  planes  travelling  with  the  velocity  of 
sound  V.  Also  let  w19  u2  be  the  speed  of  the  air  particles 
at  A! ,  A2 ;  Pi,p2  the  corresponding  pressures ;  and  d1 ,  dz  the 
densities.  Then  V—u^  and  V—  u2  are  the  velocities  of 
the  two  planes  with  respect  to  the  medium  ;'and  (  V—  HI)  dl , 
( V—  uf)  d2  are  the  masses  of  air  traversed  by  the  two 
planes  respectively  in  one  second,  since  the  cross-sectional 
area  is  one  square  centimetre.  These  masses  are  equal ; 
for  the  two  planes,  travelling  with  the  speed  of  the  sound- 
wave, remain  in  the  same  relative  position  with  respect  to 
the  condensation  or  rarefaction  of  the  wave  which  they 
accompany,  and  there  is  therefore  no  accumulation  or 
exhaustion  of  air  going  on  between  them  during  the  mo- 
tion. As  much  air  streams  in  through  AI  as  out  through 
A2.  Moreover  the  mass  of  air  traversed  per  second  is  the 
same  as  if  the  planes  were  travelling  with  a  velocity  V 
and  there  were  no  sound-wave.  We  may  accordingly 
write 

(F-M1)rfl=(r-w2)*=Fa=w.   .   .   o) 

The  change  of  momentum  of  the  mass  ?//  transferred  in 
one   second  from  one  plane  to    the  other   is   m  (w2  —  ?^), 


NATURE  AND    MOTION    OF    SOUND.  147 

But  the  rate  of  change  of  momentum  is  force,  or,  in  this 
case,  difference  of  pressure.     Therefore 


T^  /•      N  T7-  W  T7"  ^ 

From  (a),        M!  —  F—      ;  ^2  —  V . 

Therefore  u»—  ui  =  m  I  -,  —  -r  I . 

V*   <v 

Substituting  in  (£>), 


Let  the  volumes  containing  unit  mass  of  air  at  densities 
c?i,  c?2,  and  d  be  represented  by  «i,  s2,  and  s.  Then,  since 
these  volumes  are  the  reciprocals  of  the  corresponding 
densities,  we  have 

P'2  —P\  =  ™?  (*1  —  «2)  =    V*^  (S:  —  82). 

Whence  V*  =  s2  ^^l. 


Let  e  be  the  coefficient  of  elasticity  of  the  air.  It  is 
the  quotient  of  the  stress  by  the  strain,  or  the  quotient  of 
the  applied  pressure  by  the  voluminal  compression  pro- 
duced. But  the  pressure  producing  the  compression  is 
p2—pi,  and  the  compression  in  volume  is  the  diminution 
in  volume  81  —  s2  divided  by  the  original  volume  s.  There- 
fore 


Then  Fa  = 

i 

and  F=-X/J. 


148  SOUND. 

118.  Elasticity  equals  Pressure.  —  Let  P  and  d  rep- 
resent the  corresponding  pressure  and  density  of  the  air. 
Let  the  pressure  be  increased  by  a  small  quantity  p  and 
let  the  volume  of  unit  mass  of  air  be  diminished  in  conse- 
quence by  a  small  quantity  s,  the  volume  at  pressure  P 
being  8.  Then  by  Boyle's  law,  the  temperature  remaining 
constant, 

P:P  +  p::S-8:S. 

By  subtraction 

p  :  P  +  p  :  :  s  :  S, 

or  -=J-  —  —  —  ,  the  compression. 

Therefore  the  coefficient  of  elasticity  for  isothermal  com- 
pression is 


If  the  disturbance  is  such  as  to  make  a  relatively  small 
change  in  density,  p  is  negligible  in  comparison  with  P, 
and  the  coefficient  of  the  elasticity  of  volume  is  equal  to 
the  pressure  to  which  the  gas  is  subjected.  Then 


d 

But  if  the  disturbances  are  violent,  p  +  P  is  no  longer 
sensibly  equal  to  P.  The  velocity  for  violent  explosions 
is  greater  than  for  moderate  sounds.  Many  observations 
confirm  this  conclusion.  Captain  Parry  relates  that  in  the 
arctic  regions  the  report  of  a  gun  in  artillery  practice  was 
often  heard  by  a  distant  observer  before  the  command  to 
fire.  Regnault  found  that  the  velocity  of  sound  diminishes 
as  the  distance  from  the  source  increases.  The  same  con- 
clusion was  reached  by  Stone. 

Regnault  also  concluded  that  for  musical  sounds  per- 


NATURE    AND    MOTION    OF    SOUND.  149 

ceived  by  the  ear  the  apparent  velocity  of  acute  sounds  is 
sensibly  less  than  of  grave  ones.  But  the  observations  are 
complicated  by  the  fact  that  the  sensation  of  hearing  is 
excited  more  promptly  by  grave  notes  than  by  acute  ones. 
The  consequence  is  that  when  a  sound  travels  through  a 
great  length  of  conduit  it  changes  its  quality  or  timbre. 

It  is  a  fact  of  common  observation,  however,  that  within 
moderate  limits  the  velocity  of  sound  is  independent  of 
pitch  and  loudness.  If  this  were  not  so,  then  music, 
played  by  several  instruments  at  a  distance,  would  reach 
the  listener  out  of  time,  and  hence  confused  and  discordant. 

119.  Newton's  Form  of  the  Equation  for  Velocity.  - 
From  Art.  102,  P  =  Hdg,  where  H  is  the  height  of  the 
homogeneous  atmosphere. 

Therefore      ?-  =  gH,  and  V  =  A  /-  = 

d  \l  d 

If  a  heavy  body  fall  in  a  vacuum  through  a  height  H, 
the  velocity  attained  is  v  =  */*2(/H. 

The  speed  of  sound  is  therefore  equal  to  the  velocity 
acquired  by  a  body  falling  in  a  vacuum  through  half  the 
height  of  the  homogeneous  atmosphere.  It  was  in  this 
form  that  Newton  announced  the  result  of  his  investigation. 

But  'H  =  7.99  x  105  cms.  and  g  ==  980. 

Therefore   V  =  \/980  x  7.99  x  105  =  27,972  cms. 

This  is  only  84  per  cent  of  the  observed  velocity. 

120.  Corrections  for  Temperature  (V..,  II,  52).  —  The 
effects  on  the  velocity  of  sound  in  air,  due  to  changes  in 
temperature  arising  from  two  distinct  causes,  must  be  dis- 
tinguished from  each  other.     One  change  is  that  of  the 
average  temperature  of  the  air  through  which  the  sound 


150  SOUND. 

passes  ;  the  other  is  that  arising  from  compression  and 
rarefaction  in  the  two  complementary  portions  of  a  sound- 
wave. The  former  may  be  considered  as  affecting  only 
the  pressure  P  ;  the  latter  augments  the  elasticity  inde- 
pendently of  pressure. 

First,  consider  the  effect  of  a  change  of  temperature 
upon  pressure. 

Let  t  represent  the  temperature  and  a  the  coefficient  of 
expansion  of  a  gas.  The  value  of  a  is  0.003665. 

The  expression  for  velocity  becomes  then 


Vt  =  F0  */l+=  VQ  \/l  +  0.003665*. 
The  increase  in  velocity  for  one  degree  C.  is  therefore 


F0  VI  +  0.003665  -  VQ=  0.00183  F0. 

Taking  F0  as  332.4  metres,  the  increase  per  degree  C.  is 
332.4  x  0.00183  =  0.608  metre,  or  23.9  inches. 

Second.  The  value  of  the  velocity  of  sound  calculated 
from  Newton's  formula  is  J  less  than  that  furnished  by 
experiment:  The  cause  of  this  disagreement  was  dis- 
covered by  Laplace  in  1816. 

The  coefficient  of  elasticity  equals  the  pressure  applied 
to  the  gas  only  under  the  condition  applying  to  Boyle's 
law  —  that  .  the  temperature  remain  constant.  In  other 
words,  it  is  the  coefficient  applicable  to  isothermal  expan- 
sion or  condensation  when  slow  changes  take  place  under 
a  long-continued  stress.  But  Laplace  observed  that  by 
reason  of  the  poor  conductivity  and  radiating  power  of 
gases  and  the  rapidity  of  the  transmission  of  sound,  the 
heat  developed  in  any  layer  by  the  condensation  could  not 
immediately  distribute  itself  throughout  the  entire  mass  ; 
and  that  one  should  not,  therefore,  apply  to  it  Boyle's  law, 
which  supposes  a  constant  temperature.  If,  on  the  con- 
trary, the  heat  remains  entirely  localized  in  the  layer  where 


NATURE    AND    MOTION    OF    SOUND.  151 

it  is  produced,  the  phenomenon  is  subject  to  the  formula 
of  Poisson, 

pv^  —  constant, 

instead  of  the  formula  of  Boyle, 

pv  =  constant. 

7  is  the  ratio  of  the  specific  heat  of  a  gas  under  con- 
stant pressure  to  its  specific  heat  under  a  constant  volume. 
The  difference  is  that  the  coefficient  of  elasticity  to  be 
employed  to  bring  the  phenomenon  under  Poisson's  for- 
mula is  that  corresponding  to  expansion  or  compression 
without  the  entrance  or  escape  of  heat.  Such  expansion  or 
compression  is  called  adiabatic.  The  heat  effects  are  then 
all  localized  in  the  same  masses  of  air  where  they  are  pro- 
duced. This  latter  coefficient  of  elasticity  is  1.41  times 
the  other.  The  full  formula  then  becomes 


- 


121.  Computation  of  the  Velocity  of  Sound  in  Air. 
-  The  velocity  of  sound  in  dry  air  at  0°  C.  may  be  com- 
puted readily  by  the  help  of  the  formula. 

Pressure  P  under  standard  conditions  is  1,012,630 
dynes  per  square  centimetre  (101). 

The  density  d  under  the  same  standard  conditions  of  0° 
C.  and  76  centimetres  pressure  of  mercury  is  0.001293. 


122.     Velocity  of  Sound   in   Water  (V.,    II,   73).- 
The  general  formula 


152  SOUND. 

is   directly  applicable  ;    and   since    the   compression  of   a 
liquid  produces  no  appreciable  heating  effect, 
increase  of  pressure     _  me/  If 
compression  produced          k 
Here  g  is  the  acceleration  of  gravity, 
m  the  density  of  mercury, 

H  the  height  of  the  normal  barometric  column, 
k  the  coefficient  of  compressibility  of  the  liquid  or 

the  compression. 
A  pressure  of  one  atmosphere  produces  a  compression  k. 

v 

d  being  the  density  of  the  liquid  at  the  temperature  of  the 
observation. 

At  4°  C.,  k  for  water  is  0.0000499,  and  its  density  is 
unity. 

q,,       ,         7r         I  980  x  13.596  x  76          0  ,m 
Therefore     K=         -  =  WOO  cms., 


or  1425  metres. 

In  1827  Colladon  and  Sturm  measured  with  much  care 
the  velocity  of  sound  in  the  water  of  Lake  Geneva  between 
two  boats  anchored  at  a  distance  apart  of  13,487  metres. 
The  mean  time  required  for  the  transmission  of  the  sound 
of  a  bell  struck  under  water  was  9.4  seconds.  This  gives 
for  the  velocity  at  8°.l  C.,  1435  metres. 

This  is  in  very  close  agreement  with  the  calculated 
value.  The  uncertainty  relative  to  the  value  of  k  does 
not  permit  of  a  rigorous  comparison  between  theory  and 
experiment. 

123.  Velocity  of  Sound  in  Solids.--  The  same  gen- 
eral formula,  J^  —  *  I  —  ,  is  applicable  to  solids.  Thus  for 


NATURE    AND    MOTION    OF    SOUND.  153 

copper,  Young's  modulus  of  elasticity,  e,  is  120  xlO10,  and 
its  density  is  8.8. 

Therefore  V=^l  12QQXQ1Q1P  -  369,300    cms.,   or   3,693 

v        8.8 

metres. 

This  is  about  11.1  times  as  great  as  the  velocity  in  air. 
Wertheim  by  an  indirect  experimental  method  found  it 
11.167  times  as  great. 

For  steel  e  =  202  x  1010,  and  d  is  7.8. 


Therefore  r=x/202ryxl°10=  508,400   cms.,   or  5,084 

V        7.8 

metres. 

124.  Relations  between  Velocity,  "Wave-Length,  Vi- 
bration-Frequency, and  Period.  —  If  n  is  the  number  of 
complete  oscillations  of  an  air  particle  per  second,  that  is, 
the  vibration-frequency;  X  the  wave-length  or  the  distance 
the  sound  travels  during  a  complete  period  of  vibration 
T;  then,  since  X  is  the  length  of  one  'wave  and  n  such 
waves  proceed  from  the  source  in  one  second,  and  extend 
over  the  distance  F",  it  follows  that 

Y 

\n  —  V,  or  X  =  —  . 
n 

Also  since  the  vibration-frequency  is  the  inverse  of  the 

period,  or  T=  —  ,  we  have 
n 

x     rr-      ri^*,.  i 


125.  Simple  Harmonic  Motion  applied  to  Sound.  - 
Let  us  next  turn  our  attention  to  the  motion  of  the 
medium.  When  a  tuning-fork  is  set  vibrating  and  is  left 
to  itself  the  intensity  of  the  sound  diminishes,  but  the 
pitch  remains  constant.  Now  pitch  depends  upon  the 
vibration-frequency.  The  constancy  of  pitch  therefore 


154 


SOUND. 


indicates  constancy  of  vibration-frequency.  The  vibrations 
of  the  fork  are  isochronous,  like  those  of  the  pendulum. 

Unless  an  elastic  body,  like  a  tuning-fork  or  a  stretched 
string,  be  very  widely  distorted,  its  periodic  time,  and 
therefore  the  pitch  of  the  sound  produced  by  it,  are  inde- 
pendent of  the  amplitude  of  vibration.  Hooke's  law  of 
the  proportionality  of  the  forces  of  restitution  to  the 
distortion  is  a  fundamental  law  of  the  vibratory  motions 
which  give  rise  to  musical  sounds.  This  means  that  in  a 
tuning-fork,  for  example,  the  acceleration  is  proportional 
to  the  displacement. 

We  therefore  conclude  that  the  oscillations  of  the  parts 
of  musical  instruments,  as  well  as  the  swing  of  the  air 
particles  to  which  they  give  rise,  may  all  be  studied  as 
simple  harmonic  motions. 

126.  "Wave  Motion  as  a  Curve  of  Sines  (A.  and  B., 
356;  Everett's  Vibratory  Motion  and  Sound,  46).  —  If 
the  motions  of  the  particles  of  air  in  sound  are  simple 


ABCDEFGHIJ-KL 

Fig.  70. 

harmonic,  then  when  a  simple  fundamental  tone,  without 
admixture  with  other  higher  tones,  is  transmitted  through 
the  air,  the  relative  positions  of  the  air  particles  along 
the  line  of  transmission  may  be  made  out  graphically 
by  the  help  of  the  auxiliary  circle  employed  in  describing 
simple  harmonic  motion.  If  the  half  circles  of  Fig.  70 


NATURE     AND    MOTION    OF    SOUND.  155 

be  divided  into  the  same  number  of  equal  arcs,  and  vertical 
lines  be  drawn  through  the  equal  divisions,  their  positions 
will  represent  those  of  the  particles  of  air  as  they  are  dis- 
placed in  a  sound-wave  for  a  single  amplitude  of  vibration. 
At  B,  D,  F,  etc.,  they  are  crowded  together  as  a  condensa- 
tion ;  at  A,  (7,  E,  etc.,  they  are  separated  from  one  another 
as  a  rarefaction.  The  distances  BD,  DF,  etc.,  are  all  the 
same  and  equal  to  one  wave-length  of  the  sound  in  air. 

But  for  purposes  of  graphical  illustration  it  may  be  sup- 
posed that  the  vibrations  are  at  right  angles  to  the  motion 
of  the  wave,  since  in  no  other  way  can  a  curve  be  made  to 
do  duty  in  exhibiting  the  phenomena  of  wave  motion  in 
sound.  A  sinusoidal  curve  may  then  be  constructed 
to  represent  either  the  displacements  of  successive  particles 
or  their  velocities  at  a  given  instant.  For  this  purpose  a 
uniform  rectilinear  motion  is  combined  with  a  simple 
harmonic  motion  at  right  angles  to  it. 

If  a  is  the  amplitude  of  the  motion  and  y  the  displace- 
ment of  the  particle  at  the  time  £,  then  the  two  equations 
of  motion  are 


y  =  a  sin  —  -,  (Art.  33) 

x  —  vt. 

In  this  second  equation  v  is  the  velocity  of  the  uniform 
motion.     The   value    of   t   from   the   second   equation   is 

-.     Substitute  in  the  first  equation,  and 


y  =  a  sin  -—  =  a  sm  — - . 

J.  V  A, 

This  equation  shows  that  y  is  a  periodic  function  of  re, 
since  the  sine  of  an  angle  increasing  with  the  time  has 
regularly  recurring  values.  The  same  value  of  y  recurs 
with  every  increase  of  x  equal  to  X,  the  wave-length. 


156 


SOUND. 


If  various  values  are  given  to  x  the  corresponding  values 
of  y  will  represent  the  displacements  of  the  particle  the 
distance  of  which  from  the  origin  is  x.  For  x  —  0,  y  =  0 ; 
for  x  —  i\,  y  --  a ;  for  x  ==  £X,  #  =  0  ;  for  a  =  IJX, 
y  —  —  a  ;  and  for  a;  =  X,  y  =  0.  Laying  off  the  several 
values  of  x  along  a  straight  line,  and  erecting  perpen- 
diculars equal  to  the  corresponding  values  of  y,  the  curve 
drawn  through  the  extremities  of  all  the  ordinates  is  a 
curve  of  sines  or  a  sinusoid. 

Similarly  the  formula  for  the  curve  representing  veloci- 
ties  is  y  =  2™  cog  2w* 

From  the  two  equations  it  is  evident  that  the  maximum 
velocity  of  a  particle  occurs  simultaneously  with  minimum 

displacement  and  vice  versa. 

Of  the  two  curves  in  Fig. 
71  the  ordinates  of  the 
upper  one  represent  the  dis- 
placements of  a  particle  at  suc- 
cessive equal  time  intervals, 


Fig.  71. 


and  those  of  the  lower  one,  the  corresponding  velocities. 

A  curve  of  sines  may  be  drawn  by  drawing  a  circle  with 
a  radius  equal  to  the  amplitude  of  vibration  and  dividing 


ABCDEFGHIJKLMNOPQRSTUVWXYZ 


C"  —   1  ^~-*                  p 

j; 

/ 

\ 

7 

V 

/'        '\ 

'' 

\ 

/ 

\ 

/ 

\ 

5- 

I 

\ 

/ 

\ 

1 

\ 

felpgj  - 

C 

\ 

/ 

\ 

1 

j 

\ 

1 

\ 

1 

U^^=-  ?-r=^»                  1 

,^L 

J. 

X 

-'• 

f_ 

I 

\ 

I 

ABCDEF 


Fig.  72. 


it  into  a  number  of  equal  arcs,  which  shall  be  some  multiple 
of  four  —  sixteen  for  example,  as  in  Fig.  72.     Through 


NATURE    AND    MOTION    OF    SOUND.  157 

these  equal  divisions  draw  the  horizontal  parallel  lines, 
and  lay  off  along  the  line  through  0  equal  divisions  to 
represent  the  distances  x  in  the  equal  time  intervals  re- 
quired for  the  point  in  the  auxiliary  circle  to  pass  from 
1  to  2,  2  to  3,  3  to  4,  etc.  Then  erect  perpendiculars 
through  the  equal  divisions  of  Oe  produced,  and  draw  a 
plane  curve  through  the  intersections  A,  6,  c,  t?,  £,/,  etc. 
This  curve  will  be  a  sine  curve.  The  ordinates  are  pro- 
portional to  the  sines  of  an  angle  varying  with  the  time. 

Such  a  curve  may  be  drawn  experimentally  by.  securely 
clamping  one  end  of  a  long  slender  strip  of  clear  wood  so 
that  it  can  vibrate  in  a  horizontal  plane  near  a  table  top. 
To  the  free  end  attach  a  small  camel's-hair  brush,  just 
touching  a  strip  of  paper  attached  to  a  board  below  it. 
Ink  the  brush  and  vibrate  the  wood  strip;  at  the  same 
time  draw  the  board  with  the  attached  paper  along  the 
table  under  the  brush.  It  will  trace  a  sine  curve  if  the 
motion  of  the  paper  is  uniform. 

In  a  similar  manner  a  large  tuning-fork  may  be  made  to 
inscribe  its  vibrations  on  smoked  paper  fastened  round  a 
drum,  which  can  be  rotated  with  a  uniform  angular  mo- 
tion, while  a  light  tracing-point  attached  to  the  fork 
inscribes  a  sine  curve. 

127.  Composition  of  Simple  Harmonic  Motions  in 
the  Same  Plane  (A.  and  B.,  359).  -  -  The  curve  of  sines 
may  be  used  to  illustrate  the  composition  of  two  or  more 
wave  motions  in  the  same  plane.  If  two  systems  of  waves 
coexist  in  the  same  medium  the  displacement  at  any  point 
will  be  the  sum  of  the  displacements  due  to  the  two  sys- 
tems taken  separately.  Hence  the  actual  displacements 
may  be  found  by  taking  the  algebraic  sum  of  the  ordinates 
of  the  two  displacement  curves.  If  the  two  systems  have 


158 


SOUND. 


Fig.  73. 


the  same  period,  then  the  resulting  curve  will  be  a  simple 

sine    curve    of    the    same 

period;  if  the  periods  are 

not  the  same,  the  composite 

curve  will  not  be  a  curve 

of  sines. 

In   Fig.    73   the    dotted 

and    light   lines   represent 

the  displacements   due    to 

two   wave   systems  of  the 

same  period  and  amplitude. 

The  heavy  line  represents 

the  resulting  displacement,  in 

In  /the  two  systems  have 

the  same  phase,  and  the  re- 

sulting amplitude  is  double 

that  of  either  component.     In  //  the  phases  differ  by  one- 

quarter,  and  in  III  by  one-half  a  period.     In  the  last  case 

the  two  motions 
completely  annul 
each  other.  In  Fig. 
74  the  periods  of 
the  two  wave  sys- 
tems are  as  1  to 
2.  The  resulting 
curve  is  not  a  sinus- 
oid, whether  the 
component  waves 
are 

phase, 

upper  part  of  the 
figure,  or  not.  The 

composite  curve  is  periodic,  however  ;  that  is,  it  has  values 

of  the  ordinates  recurring  at  equal  time  intervals. 


in    the    same 
as    in    the 


Fig.  74. 


NATURE    AND    MOTION    OF    SOUND.  159 

128.  Interference  and  Beats  (V.,  II,  92). — The  last 
topic  illustrates  the  superposition  of  two  wave  systems  in 
the  same  medium.  In  the  case  of  two  water-waves  the 
total  elevation  or  depression,  relative  to  the  primitive  level, 
is  at  each  point  and  at  each  instant  equal  to  the  algebraic 
sum  of  the  displacements  due  to  each  system  separately. 
If  an  elevation  of  the  first  system  is  superposed  on  an 
equal  elevation  of  the  second,  the  total  height  of  the  water 
above  its  primitive  level  will  be  double  the  elevation  due 
to  one  system  alone.  If  an  elevation  of  the  first  system 
coincides  with  an  equal  and  opposite  depression  of  the 
second,  the  primitive  level  will  not  be  modified. 

In  sound-waves  condensations  and  rarefactions  take  the 
place  of  elevations  and  depressions,  but  two  sound-wave 
systems  modify  each  other  in  much  the  same  manner  as 
two  systems  of  water-waves.  Consider  in  particular  two 
identical  sources  of  sound  A  and  B.  It  is  clear  that  at 
every  point  of  a  plane,  drawn  perpendicular  to  AB  at  its 
middle  point,  the  movements  provoked  by  the  two  systems 
of  waves  at  each  instant  will  be  concordant,  or  will  reen- 
force  each  other.  The  velocity,  as  well  as  the  displace- 
ment, will  be  greater  than  if  there  were  only  a  single 
source  of  sound. 

But  in  any  other  plane,  parallel  for  example  to  AB,  the 
velocity,  and  in  consequence  the  intensity,  will  present 
a  series  of  fixed  maxima  and  minima.  So  two  sounds 
emanating  from  two  identical  centres  reenforce  each  other 
at  certain  points  of  space,  and  destroy  each  other  at  other 
points,  or  sound  added  to  sound  produces  silence.  This 
phenomenon  is  called  interference. 

When  the  two  component  sound-waves  have  nearly  the 
same  period  the  case  deserves  special  attention.  The  re- 
ciprocal action  results  in  periodic  interference  and  beats. 


160  SOUND. 


For  a  few  vibrations  the  periods  may  be  regarded  as  the 
same,  and  the  resulting  vibration  will  be  simple  harmonic. 
But  the  more  rapid  vibration  will  gain  on  the  other,  thus 
changing  the  difference  of  phase  on  which  the  resultant 
depends.  For  when  the  two  systems  have  the  same  phase 
they  reenforce  each  other ;  when  they  have  opposite 
phases  they  partially  or  wholly  annul  each  other. 

Let  the  two  systems,  of  nearly  the  same  amplitude,  have 
vibration-frequencies  m  and  n,  m  —  n  being  very  small. 
Suppose  the  phases  to  agree  initially.  Then  after  an  in- 
terval in  seconds  of — the  two  systems  will  be  in 

2  (w-'/O 

exactly  opposite  phase,  one  system  having  gained  half  a 
wave-length  on  the  other,  and  almost  total  extinction  of 
motion  and  of  sound  will  ensue. 

After  a  further  interval  of  -   x  seconds,  the  system 

2(m  —  n) 

of  shorter  period  will  have  gained  a  complete  wave-length 
on  the  other,  the  two  systems  will  again  be  in  agreement 
in  phase,  and  an  increase  of  sound  will  result.  This 
phenomenon,  due  to  interference,  is  known  as  beats.  The 
number  of  times  per  second  that  the  two  systems  reenforce 
each  other  is  equal  to  the  difference  in  the  vibration-fre- 
quencies of  the  two  notes,  or  m— n.  Thus  if  m  is  103 
and  n  is  100,  then  the  first  reenforcement  will  occur  at  the 
end  of  one-third  of  a  second,  a  second  one  at  two-thirds 
of  a  second,  and  a  third  at  the  end  of  the  first  second. 

129.  Experiments  illustrating  Interference  of  Sound- 
Waves.  —  Take  two  tuning-forks  in  unison,  mounted  on 
resonant  boxes.  Stick  a  small  mass  of  wax  to  the  prong 
of  one  of  them.  This  increases  its  moment  of  inertia, 
and  so  increases  its  periodic  lime  of  vibration.  If  the  two 


NATURE    AND    MOTION    OF    SOUND.  161 

forks- are  now  sounded  together,  the  phenomenon  of  beats 
will  be  very  pronounced. 

Mount  two  organ  pipes  of  the  same  pitch  on  a  bellows, 
and  sound  together.  If  they  are  open  pipes,  a  card 
gradually  slipped  over  the  open  end  of  one  of  them  will 
change  its  pitch  enough  to  bring  out  strong  beats.  The 
same  result  may  be  produced  by  slowly  sliding  the  finger 
across  the  embouchure  of  one  pipe. 

These  two  experiments  illustrate  interference  of  two 
sounds  of  slightly  different  pitch.  The  two  sources  are 
not  identical.  The  following  one  will  serve  to  illustrate 
interference  from  two  identical  sources,  viz.,  the  two 
prongs  of  the  same  fork. 

Let  a  and  b  be  the  prongs   of  a  diapason  (Fig.  75). 
It  is  well  known  that  the  intensity  of  sound  of  a  tuning- 
fork  held  freely  in  the  hand  and  turned  on  its  stem  ex- 
hibits  changes.     As   the 
two  branches  approach  or       9  -\^ 
recede    from    each   other  \x  / 

the  movements  communi-  \x       d 

cated   to   the  air  are  all  \  / 

the   time   opposing    each  c  l(&         e 

other.  j 

While  the  two  branches,  /'        . 

for    example,     approach  /'''  \ 

each   other,    a    condensa-     ^.^  ^Nfe 

tion  is  produced  between  ng.  75. 

them,   and   at  the    same 

time  two  rarefactions  start  from  the  backs  of  the  branches 
a  and  b.  When  such  a  diapason  is  held  before  the  ear,  or 
is  placed  over  a  jar  serving  as  a  resonator,  the  sound  is 
perceived  to  be  strong  in  the  regions  d  and/,  less  intense 
at  c  and  e,  and  to  disappear  completely  on  the  branches  of 


162 

the  hyperbola  gah  and  M.  If  then  one  turns  the  fork  on 
its  stem  one  perceives  a  succession  of  reinforcements  and 
enfeeblements  of  sound.  When  the  fork  is  in  a  position 
of  least  intensity  of  sound,  the  covering  of  one  branch  by 
a  wooden  or  a  pasteboard  tube  without  touching  serves 
to  restore  the  sound  to  nearly  maximum  intensity. 

ISO.  To  combine  Two  Simple  Harmonic  Motions  at 
Bight  Angles.  —  In  this  case  we  seek  the  resultant  motion 
arising  from  impressing  simultaneously  upon  a  particle  two 
simple  harmonic  motions  at  right  angles  to  each  other. 
Each  of  the  component  motions  may  be  regarded  in  the 
usual  way  as  the  apparent  motion  of  a  point  moving  uni- 
formly around  a  circle.  Let  the  radii  of  the  two  circles  of 
reference  be  a  and  b.  The  periods  of  the  two  harmonic 
motions  may  have  any  ratio  to  each  other,  although  only  a 
few  of  the 'simpler  ratios  have  been  investigated. 

Making  use  of  the  method  adopted  in  Art.  33,  the  dis- 
placements in  the  two  rectangular  directions,  if  the  periods 
are  equal,  will  be 

x  —a  sin  (0  —  e), 

y=  5  sin  (0  —  e'). 

These  two  harmonic  motions  are  entirely  independent, 
but  they  are  to  be  impressed  upon  the  same  particle.  If 
e'  —  e  =  B ,  the  difference  of  phase,  we  can  express  the  two 
displacements  under  the  form 

x  =  a  sin  (0  +  S), 

y  —  b  sin  #, 

where  8  is  positive  if  the  x  component  is  in  advance  and 
negative  if  it  is  behind. 

Finally,  if  n  and  m  represent  respectively  the  least 
numbers  of  complete  oscillations  executed  by  the  moving 
point  in  the  two  directions  in  the  same  interval  of  time, 


NATURE    AND    M®  't'l  ON    OF    SOUND.  163 


then  we  have,  as  the  most  general  form  of  the  equations  of 
displacement, 

x  =  a  sin  (n6  +  £), 

y  —  b  sin  m6. 

In.  order  to  obtain  the  equation  of  the  path  of  the 
moving  point  referred  to  rectangular  axes,  it  is  only  neces- 
sary to  eliminate  the  angle  6  from  the  two  equations  and 
we  have  the  "  curve  of  impression  as  perceived  by  the 
eye." 

131.  To  combine  Two  Simple  Harmonic  Motions  of 
the  Same  Period  at  Bight  Angles  --  For  this  case  n  —  m 
and  the  displacements  are 

x  =  a  sin  (6  +  S), 
y—'b  sin  0. 
Expanding  the  first  equation 

x=  a  (sin  6  cos  £  +  cos  0  sin  £). 

From  the  second  equation  sin  0  =  |-  .    Substitute  in  the 

o 

equation  for  x  and 


or 


x=  I  G/  cos  fi  +  V62-/  sin  8), 
o 

I  ay  A2        °?  STA  2\       •     2    S> 

(  x  — ~  cos  6  J  =  —  (62  —  y2)  sin2  8. 
\         b  /   .  P 


This  is  an  equation  of  the  second  degree  ;  and,  since  the 
curve  returns  into  itself,  or  is  a  closed  curve,  it  must  be  an 
ellipse.  Consider  four  cases. 

(1.)  When  S  =  0  or  there  is  110  difference  of  phase 
between  the  two  component  motions.  Then  cos  8  =  1; 
sin  S  =  0.  Consequently 

ay      A  b 

x  —  -2.  =  0  or  y  —  -  x. 
b  a 


164  SOUND. 

This    is  the   equation    of   a   straight   line   through  the 
origin,  the  inclination  to  the  axis  of  x  being  tan      (-)• 

w 

(2.)     When  8  =  ^-  or  the  phase  difference  is  a  quarter 
of  a  period.     Then  cos  8  =  0;  sin  8  =  1.     Therefore 


This  is  the  equation  of  an  ellipse  referred  to  its  centre 
as  origin  and  of  semi-axes  a  and  b. 

(3.)  When  8  =  TT,  or  the  phase  difference  is  half  a 
period.  Then  cos  8  =  —  1  ;  sin  8  =  0.  Therefore 


,   a  b 

T>y  =        or  y  =  —  -x. 


This  is  also  an  equation  of  a  straight  line  through  the 

—  V    ^\ 

origin,  the  inclination  to  the  x  axis  being  tan     /  --  J  . 

Q 

(4.)     When  8  =  -TT  or  the  difference  of  phase  is  f  of  a 
period.     Then  cos  8  =  0;  sin  8  =  —  1.     Therefore 


This  is  the  equation  of  the  same  ellipse  as  in  case  (2), 
but  traced  by  the  moving  point  in  the  opposite  direction. 

All  the  ellipses  that  can  be  obtained  by  varying  8  will 
lie  within  a  rectangle  the  sides  of  which  are  2#  and  2b  ; 
for  if  x  be  made  zero  in  the  general  equation  above  con- 
necting x  and  #,  y  will  equal  ±  b  ;  while  if  y  be  made  zero, 
x  will  be  +  a.  The  ellipses  are  always  tangent  to  the  sides 
of  the  rectangle. 


NATURE    AND    MOTION    OF    SOUND.  165 

132.  To  combine  Two  Simple  Harmonic  Motions  at 
Right  Angles  with  Periods  as  One  to  Two.  —  If  we  as- 
sume the  period  of  the  x  component  to  be  one-half  that  of 
the  y  component,  then  n  =  2,  m  —  1,  and  the  two  equa- 
tions of  displacement  become 

x  =  a  sin  (20  -f-  8), 
y=b  sin  6. 

The  elimination  of  6  from  these  equations  gives  in  gen- 
eral an  equation  of  the  fourth  degree,  of  which  the  three 
following  cases  are  of  most  interest  : 

(1.)    'When  8  =  0  or  TT,  sin  8  =  0.     Then 

sin  0=;  ?=  sin  2(9  =  2  sin  6  cos0. 


But  if  sin  0  =  ^  ,  cos  0  = 


Therefore 


* 

or  lW 

This   is   the    equation   of    a   lemniscate    symmetrically 
placed  on  the  y  axis. 

(2.)     When  8  =  -.     Then  cos  B  =  0  ;  sin  8  =  1. 

Zi 

Therefore  x  =  a  (sin  20  cos  8  +  cos  20  sin  8), 

or  x  =  acos20=a  (2  cos2  0  -  1). 

Substitute  now  the  value  of  cos  0  from  (1)  and  after 
reduction  2       b2  , 

y=^a~x^ 

This  is  the  equation  of  a  parabola  symmetrically  placed 
on  the  x  axis,  whose  vertex  is  at  a  distance  a  from   the 

origin,  and  focus  at  the  distance  ^-  from  the  vertex. 

oa 


166 


SOUND. 


3 

(3.)     When  S  =  -  IT.     Then  cos  8  =  0  ;  sin  8  = 

The  equation  then  becomes 


—  1. 


denoting   the   same    parabola   reversed,  its   vertex   being 
now  turned  in  the  direction  of  the  negative  axis  of  x. 

133.  Graphical  Method  of  combining  Two  Simple 
Harmonic  Motions  at  Right  Angles.  —  Draw  two  con- 
centric circles  (Fig.  76)  with  radii  proportional  to  the 


NATURE    AND    MOTION    OF    SOUND.  167 

amplitudes,  a  and  £>,  of  the  two  harmonic  motions,  and 
through  their  common  centre  0  draw  the  rectangular 
diameters  AB,  CD. 

Divide  each  quadrant  of  both  circles  into  the  same 
number  of  equal  parts ;  some  multiple  of  four  is  usually 
most  convenient.  Through  the  points  of  division  of  the 
circle  AB  draw  lines  parallel  to  CD,  and  through  the  di- 
visions of  CD  draw  lines  parallel  to  AB.  The  resulting 
rectangle  of  sides  2a  and  25  will  contain  all  the  figures 
arising  from  any  possible  combination  of  two  simple  har- 
monic motions  of  commensurable  periods ;  and  the  curves 
will,  in  general,  be  tangent  to  the  sides  of  the  rectangle. 
The  centre  of  the  circles  corresponds  to  a  phase  difference 
of  zero  between  the  two  components,  that  is,  to  8  =  0;  and 
it  is  taken  as  the  starting  point  for  tracing  all  curves  of 
phase  difference  zero  or  TT. 

If,  as  in  Fig.  76,  the  circles  have  been  divided  into  six- 
teen equal  parts,  then  each  point  of  intersection  on  the 

diameter  AB  corresponds  to  a  phase  difference  of  -  -  or 

16 

5- ,  that  is,  to  one-sixteenth  of  a  period.  Hence  if  we  start 
o 

to  trace  a  curve  from  a  in  the  figure  instead  of  from  0,  we 
shall  produce  the  curve  corresponding  to  a  phase  differ- 
ence of  — .  This  means  that  at  the  instant  when  the  y 

component  passes  through  AB  in  the  positive  direction, 
and  the  y  displacement  is  therefore  zero,  the  x  component 
has  already  reached  a  in  the  positive  direction,  or  is  in  ad- 
vance of  the  y  component  by  Oa  or  — .  In  like  manner  b 

o 

corresponds  to  a  phase  difference  of  j,  c  to  ~,  and  A  to 

4  8 


168  SOUND. 

•p.     Returning  toward  0,  it  will  be  seen  that  c  also  corre- 
sponds to  a  difference  of  phase  of  -5-,  b  to  -j-,  a  to  -5-,  and 

o  4  o 

0  to  TT,  with  larger  values  for  points  to  the  left  of  0. 

Suppose  now  that  we  wish  to  trace  the  curve  corre- 
sponding to  the  vibration-frequencies  one  to  two,  two  for 
the  horizontal  and  one  for  the  vertical  component,  and 
with  no  difference  of  phase.  Starting  from  0  we  count 
two  points  horizontally  to  the  right  and  one  up  and  reach 
I ;  again  two  to  the  right  and  one  up  for  point  II ;  and  so 
continue,  numbering  the  points  in  order  until  we  pass 
through  the  starting  point  in  the  same  direction  as  at  first, 
being  careful  always  to  complete  the  motion  in  one  direc- 
tion before  beginning  the  retrograde  motion.  An  excellent 
check  upon  the  accuracy  of  the  location  of  the  points  is 
found  in  the  fact  that  points  equidistant  from  the  axis  of 
symmetry  AB  differ  in  number  by  eight  in  every  case, 
that  is,  by  half  a  vibration. 

If  now  a  smooth  curve  be  traced  through  the  points  in 
order  we  see  that,  in  accordance  with  Art.  22,  the  moving 
point,  being  subject  to  both  motions,  describes  two  spaces 
horizontally  and  one  vertically  in  the  same  interval  of 
time,  and  consequently  passes  through  the  corners  of  rec- 
tangles two  spaces  long  and  one  space  high  in  every  case. 
The  spaces  themselves  increase  or  decrease  according  to 
the  simple  harmonic  law.  Great  diversity  of  figure  may 
thus  be  obtained  with  successive  differences  of  phase  be- 
tween the  two  component  motions. 

To  combine  two  motions  of  frequencies  two  to  three,  we 
should  simply  count  three  spaces  in  one  direction  and  two 
in  the  other  and  proceed  in  other  respects  as  already 
described. 


NATURE    AND    MOTION    OF    SOUND. 


169 


The  curves  obtained  in  this  way  are  most  beautifully 
verified  experimentally  by  Blackburn's  pendulum,  with 
adjustable  periods  of  vibration  in  two  planes  at  right 
angles  ;  or  by  Lissajous'  optical  method,  in  which  a  beam 
of  light  is  successively  reflected  from  mirrors  on  two 
tuning-forks,  one  vibrating  horizontally  and  the  other 
vertically. 

134.    The  Principle  of  Huyghens  (D.,  HO  ;  P.,  52 ;  L., 

229  ;  A.  and  B.,  356).  —Let  a  (Fig.  77)  be  a  centre  of 

disturbance,  and  men   the    front    of    a 

spherical  wave  diverging  from  it.     The 

radius  of  the  wave   increases  with  the 

velocity  of  sound,  and  the  disturbance 

now  at  men  will  a  moment  later  be  at 

m'dn'.     This  single  wave    as  it  travels 

outward  will  disturb  all  the  elements  of 

the  medium  over  which  it  passes.     The 

disturbance  of  any  one  element  of  the 

medium  may  then  be  considered  as  the 

cause  of  the  subsequent  disturbance  of 

all  the  other  elements.     The  principle 

of  Huyghens  is  that  every  point  of  the 

wave-surface  mn  becomes  a  new  centre 

of  disturbance  from  which  waves  of  sound  a"re  propagated 

outwards  in  the  same  manner  as  from  the  original  centre ; 

and  the  aggregate  effect  at  any  point  outside  this  surface 

is  the  resultant  of  the  combined  action  of  all  the  secondary 

waves  propagated  from  these  new  centres.     The  principle 

follows  from  the  consideration  that  every  particle  on  the 

wave-surface  has  the  same  oscillatory  motion,  except  in 

point  of  amplitude,  as  the  first  particle  disturbed ;  and  it 

therefore  stands  in  the  same  relation  to  adjacent  particles. 


Fig.  77. 


170  SOUND. 

and  communicates  motion  to  them  in  the  same  manner,  or 
becomes  itself  a  centre  of  disturbance. 

The  principle  of  Huyghens  is  the  principle  of  superposi- 
tion in  a  generalized  form.  The  disturbance  at  any  point 
at  any  instant  is  due  to  the  superposition  of  all  the  dis- 
turbances which  reach  it  at  that  instant  from  the  various 
parts  of  the  surrounding  medium. 

Let  the  points  of  the  surface  mn  be  centres  from  which 
waves  proceed  for  a  short  distance  cd.  Then  with  these 
centres  and  a  radius  <%?,  describe  semi-circular  waves.  The 
number  of  such  waves  being  indefinitely  large,  they  will 
ultimately  coalesce  to  form  the  new  surface  wV,  which 
is  the  envelope  of  all  the  small  secondary  waves.  The 
effective  part  of  each  secondary  wave  Huyghens  supposed 
confined  to  that  portion  which  touches  the  envelope. 

The  energy  of  mn  is  thus  passed  on  to  wV,  and  in  the 
same  manner  from  m'n'  to  m"n",  etc. 

But  the  question  arises,  will  not  the  disturbance  be  prop- 
agated backwards  as  well  as  forwards  by  these  secondary 
wavelets?  The  answer  is  that  each  secondary  wave  is 
limited  in  the  same  manner  as  the  primary  wave,  or  the 
agitation  of  any  point,  like  a  pulse  on  a  stretched  cord, 
causes  the  agitation  of  points  in  advance  of  it,  but  of  none 
in  the  direction  from  which  it  has  come.  The  law  of 
intensity  at  ea*ch  point  of  a  secondary  wave  has  been  in- 
vestigated by  Stokes,  who  has  shown  that  the  effect  of  an 
elementary  wave  at  any  external  point  varies  as  (1  +  cos  #), 
where  6  is  the  angle  between  the  normal  to  the  primary 
wave  and  the  line  joining  the  point  to  the  centre  of  the 
elementary  wave.  This  quantity  vanishes  when  O  =  TT^  or 
for  points  directly  behind  the  wave.  The  disturbance  due 
to  a  secondary  wave,  therefore,  varies  from  a  maximum  at 
its  forward  apex  to  zero  at  the  opposite  point  in  its  rear. 


NATURE    AND    MOTION    OF    SOUND. 


171 


135.  Reflection  of  a  Plane  Wave  at  a  Plane  Surface 
(D.,  117).  —  Let  AB  (Fig.  78)  be  a  portion  of  the  plane 
advancing  wave,  and  let  CD  be  the  reflecting  surface.  If 
AB  had  met  with  no  obstruction  it  would  have  taken  the 
position  A'B'  at 
the  instant  when 
B  arrives  at  Bf. 
But  A  becomes 
a  new  centre  of 
disturbance 
which  travels 
backward  in  the 
first  medium. 
Then  with  A  as 
a  centre,  and 
with  a  radius 
AA,  equal  to  -""^  pjg  IQ 

BB',  describe  a 

circle.  This  circle  limits  the  distance  to  which  the  dis- 
turbance has  spread.  In  the  same  time  the  disturbance 
from  b  would  have  reached  b'  without  obstruction,  but  it 
travels  first  to  E,  and  is  then  reflected.  We  must,  there- 
fore, draw  about  E  a  circle  with  radius  Ebf.  In  the  same 
manner  draw  any  number  of  circles. 

Finally,  from  B'  draw  a  tangent  to  the  first  circle  ;  it 
will  touch  all  the  other  circles,  and  will  be  the  reflected 
wave-front.  Draw  A  A"  to  the  point  of  tan  gen  cy  with  the 
first  circle.  Then  AA"B'  is  symmetrical  with  respect  to 
AAB' ;  and  since  ABB' A  is  a  parallelogram,  the  triangle 
AA'B'  is  equal  to  ABB'.  Therefore,  since  the  triangles 
AA'B'  and  ABB'  are  both  equal  to  AAB',  they  are  equal 
to  each  other,  and  the  angles  BAB'  and  A"B'A  are  equal 
to  each  other.  But  the  former  is  the  angle  of  incidence, 


172 


SOUND. 


since  the  lines  BA  and  B'A  are  perpendicular  respectively 
to  BE'  and  nB',  and  the  angle  BB'n  is  the  angle  of  incidence. 
In  the  same  way  it  can  be  shown  that  the  angle  A'.B'A  is 
equal  to  nB'r,  the  angle  of  reflection.  The  angle  of  inci- 
dence, therefore,  equals  the  angle  of  reflection.  The 
former  is  the  angle  between  the  incident  wave-front  and 
the  reflecting  surface  CD;  the  latter  is  the  angle  between 
the  wave-surface  after  reflection  and  the  reflecting  sur- 
face. The  reflection  of  sound  then  follows  the  ordinary 
laws  of  the  reflection  of  waves. 


136.  Relations  of  the  Centres  of  the  Direct  and  Re- 
flected Systems  of  Waves.  —  Let  0  (Fig.  79)  be  the 
centre  of  the  incident  spherical  waves,  and  let  them  be 
reflected  from  the  surface  AB.  If  these  waves  had  met 
with  no  obstruction  they  would  have  taken  positions  at 
equal  successive  time-intervals  indicated  by  the  dotted 
lines  ;  but  they  are  reflected  so  as  to  have  the  positions  of 

the  full  lines  sym- 
metrically situated 
on  the  other  side  of 
the  reflecting  sur- 
face. Let  01  be  a 
sound-ray,  or  the 
direction  of  motion 
of  any  point  of  the 
incident  wave ;  draw 
JIT  so  that  IM  and 
O/ shall  make  equal 

angles  with  the  normal.  Then  is  IM  the  path  of  the 
reflected  ray ;  that  is,  it  is  a  normal  to  the  reflected  waves. 
Project  IM  backward  till  it  intersects  at  0'  the  normal  to 
the  reflecting  surface  through  0.  Then  is  0'  the  centre 


NATURE    AND  MOTION    OF   SOUND.  173 

of  the  reflected  waves.  The  triangles  QIC  and  0'IC  are 
equal.  Hence  00  and  O'(7are  equal,  and  the  centres  of 
the  incident  and  reflected  waves  are  on  a  perpendicular  to 
the  reflecting  surface  and  equidistant  from  it.  The  sound- 
centre  and  sound-image  are  symmetrically  situated  with 
respect  to  the  reflecting  surface. 

137.  Echo  (D.,  425) Whenever  sound  passes  from 

one  medium  to  another  of  different  density  a  part  of  the 
energy  is  transmitted  and  a  part  reflected.  The  one  system 
of  waves  gives  rise  to  two  distinct  systems,  and  the  inten- 
sity of  sound  in  either  direction  is  weakened  by  the  new 
medium.  In  general  the  energy  or  intensity  of  the  re- 
flected system  increases  with  the  difference  in  density  of 
the  two  media.  A  dry  sail,  for  example,  will  transmit  a 
part  of  the  sound,  and  will  reflect  a  part;  but  if  wetted 
it  becomes  a  better  reflector,  and  almost  impervious  to 
sound. 

A  flame  is  known  to  be  a  fairly  good  reflector  of  sound, 
and  the  hot  air  above  the  flame  reflects  nearly  as  well  as 
the  flame  itself.  It  is  evident  from  this  fact  that  if  the  air 
in  clear  weather  has  ascending  and  descending  currents, 
differing  in  temperature  from  the  neighboring  masses,  the 
sound  will  be  partly  reflected  and  partly  transmitted.  Its 
intensity  will  then  fall  off  with  distance  much  more 
rapidly  than  if  the  air  were  of  uniform  density.  Sound  is 
often  heard  more  plainly  in  foggy  or  rainy  weather  than 
when  the  atmosphere  is  clear,  because  then  the  air  is  more 
uniform.  For  the  same  reason  sounds  are  heard  much 
further  in  a  quiet  night  than  by  day.  Humboldt  remarks 
on  the  great  distance  that  the  falls  of  the  Orinoco  in  South 
America  can  often  be  heard  by^night;  and  arctic  travellers 


174  SOUND. 

relate  that  in  the  long  night  of  the  polar  regions  a  slight 
sound  can  be  heard  an  incredible  distance. 

The  most  important  illustration  of  the  reflection  of 
sound-waves  occurs  in  the  case  of  echoes.  Assume  that 
the  sensation  of  sound  persists  for  about  one-tenth  of  a 
second,  during  which  time  sound  travels  33  metres. 

If  then  the  distance  of  the  reflecting  surface  exceeds  16.5 
metres,  the  observer  may  hear  the  direct  and  reflected 
sounds  separately.  This  repetition  of  sound  by  reflection 
is  called  echo.  Parallel  reflecting  surfaces  at  suitable  dis- 
tances produce  multiple  echoes.  Reflection  of  sound  takes 
place  from  buildings,  rocks,  woods,  and  hills.  If  a  person 
can  utter  five  syllables  a  second,  standing  opposite  a  large 
reflecting  surface,  at  a  distance  of  165  metres,  he  can  then 
hear  the  five  syllables  repeated  by  reflection  entirely  dis- 
tinct from  the  original  words.  For  shorter  distances  the 
the  direct  and  reflected  sounds  become  confused.  Such  is 
case  in  rooms  with  bad  acoustic  properties.  A  circular 
building  with  a  hemispherical  dome,  like  the  Baptistry  at 
Pisa,  may  prolong  a  sound  for  many  seconds  by  successive 
reflections.  The  effect  is  made  more  conspicuous  by  the 
good  reflecting  surfaces  of  polished  marble.  A  single 
loud  sound  in  the  Baptistry  at  Pisa  continues  to  be  audi- 
ble for  twelve  or  fifteen  seconds. 

In  a  similar  way  the  sound  of  a  whistle  or  a  gun  on  the 
water  is  often  heard  to  roll  away  apparently  to  a  great 
distance.  These  are  called  aerial  echoes.  A  curious  echo 
of  this  kind  was  observed  by  Tyndall  during  a  course  of 
experiments  near  Dover  to  determine  the  effectiveness 
of  sound-signals  during  a  fog.  An  atmosphere  perfectly 
transparent  from  an  optical  point  of  view  may  have  an 
acoustic  opacity  almost  impenetrable.  If  considerable 


NATURE   AND    MOTION    OF   SOUND.  175 

masses  of  invisible  vapor  rise  from  water  they  may  become 
obstacles  for  the  transmission  of  sound,  by  creating  hetero- 
geneous layers  or  banks,  at  the  limiting  surfaces  of  which 
sound  will  be  partially  reflected.  A  portion  of  the  sound 
transmitted  by  one  bank  is  then  reflected  by  the  next, 
giving  rise  to  a  curious  prolongation  of  a  short  signal. 

PROBLEMS. 

1.  Find  the  wave-length  in  air  of  a  note  due  to  128  vibrations  per 
second  when  the  temperature  is  20°  C. 

2.  An  express  train  passes  a  station  at  a  speed  of  70  kilometres 
an  hour,  and  blows  a  whistle,  the  frequency  of  which  is  750  vibrations 
per  second.     What  will  be  the  difference  in  pitch  of  the  note  to  an 
observer  at  the  station  as  the  train  approaches  and  as  it  recedes, 
temperature  20°  C.  ? 

3.  A  stone  is  dropped  into  a  well  and  is  heard  to  strike  after 
3  seconds.     Determine  the  depth,  the  velocity  of  sound  being  335 
metres  per  second. 

4.  If  the  velocity  of  sound  is  332  metres  per  second,  find  the 
number  of  vibrations  which  a  C  fork,  with  a  frequency  of  256,  will 
make  before  the  sound  is  audible  at  a  distance  of  50  metres. 

5.  Three  observers  are  stationed  2,  4,  and  6  kilometres  respect- 
ively from  a  gun,  which  is  fired  at  noon.     At  what  time  will  the 
report  be  heard  by  the  several  observers  if  there  is  no  wind  and  the 
temperature  is  20°  C.  ? 

6.  If  in  the  preceding  example  the  wind  is  blowing  at  a  speed  of 
80  kilometres  an  hour,  at  what  time  will  the  report  be  heard  by  the 
first  observer  stationed  directly  windward   and  the  second  directly 
leeward  ? 


176  SOUND. 


CHAPTER   VII. 


PHYSICAL   THEORY    OF   MUSIC. 

138.  Musical  Intervals.  —  The  pitch  of  a  musical 
sound  is  the  pitch  of  its  gravest  component,  or  funda- 
mental tonex ;  and  this  depends  upon  the  frequency  of  the 
fundamental  vibrations  of  the  sounding  body. 

Pitch  may  be  denned  in  two  ways : 

1.  Physically,  as  the  number  of  vibrations  per  second 
in  the  lowest  tone  of  the  sound. 

2.  Musically,  by  referring  the  sound  to  its  place  in  an 
arbitrary  scale  of  pitch  in  use  among  musicians. 

When  two  notes  are  sounded  together  or  in  quick 
succession,  the  ear  recognizes  a  special  relationship  exist- 
ing between  them,  involving  a  perception  of  their  relative 
pitch. 

This  relationship  is  expressed  as  a  ratio  between  their 
frequencies  of  vibration,  and  is  entirely  independent  of  the 
absolute  pitch  of  the  two  tones.  It  is  called  a  musical 
interval. 

Many  of  these  ratios  have  definite  names  in  musical 
nomenclature.  Thus  the  ratio  of  one  to  one  is  called 

2  3  34 

unison  ;  of  ^-,  an  octave  ;  of  ^-,  a  twelfth;  of  -,  a  fifth;  of  ~ ,  a 

fourth  ;  of  -r ,  a  major  third  ;  of  ^ ,  a  minor  third.     Numeri- 
cally a  musical  interval  is  always  equal  to  or  greater  than 


PHYSICAL    THEORY  OF  MUSIC.  177 

unity.     Musical  intervals  are  equal  to  each  other  when  their 
constituent  notes  have  the  same  relative  vibration-rates. 

139.  The  Diatonic  Scale  or  Gamut.  —  When  three 
notes,  whose  vibration-rates  are  as  4 :  5 :  6,  are  sounded 
together  an  effect  is  produced  which  is  pleasing  to  the  ears 
of  Western  nations,  as  distinguished  from  those  of  the 
Orient.  Such  a  combination  of  three  tones  is  called  a 
major  triad ;  and,  together  with  the  octave  of  the  lowest 
of  the  three,  they  compose  a  major  chord.  The  perfect  dia- 
tonic scale  is  constructed  on  three  sets  of  such  triads. 

If  the  three  tones  have  vibration-frequencies  as  10 :  12 : 
15,  they  compose  a  minor  triad  ;  and  with  the  octave  of  the 
lowest,  a  minor  chord. 

A  single  tracing  point,  like  the  graver  on  the  diaphragm 
of  a  phonograph,  may  be  set  in  motion  by  two  or  more 
systems  of  sound-waves  simultaneously.  If  then  the  sur- 
face on  which  the  curve  is  to  be  inscribed  is  moved  at 
right  angles  to  the  motion  of  the  tracing  point,  the  result- 
ing curve  will  be  due  to  the  superposition  of  the  several 
motions  in  the  same  plane.  The  upper  curve  in  Fig.  80 

4:5:0  f 

/lWxW\|/V-^ 

101: 12:15 


Fig.  80. 

is  the  result  of  combining  in  this  way  three  simple  harmonic 
motions  constituting  a  major  triad ;  the  lower  curve  in  the 
same  way  shows  the  composite  motion  resulting  from  a 
minor  triad.  It  will  be  noticed  that  a  complex  wave- 
form recurs  in  each  case,  but  less  frequently  in  the  second 
combination  than  in  the  first. 


178  SOUND. 

The   eight   notes  of   the    scale    are  represented  by   the 
letters  <7,  D,  E,  F,  a,  A,  B,  c. 
The  three  major  triads  are 

O:  E:  a  |  l 

a:  B:  d   I:  :  4:  5:  6. 

F:  A:    c) 

When  the  several  notes  of  the  scale  are  thus  related 
they  give  the  most  pleasing  chords. 

If  then  C  is  due  to  m  (about  64)  vibrations  per  second, 
the  vibration-rates  of  the  other  notes  of  the  scale  may  be 
found  by  simple  proportion  from  the  above  relations. 

XT          £  E  C 

mi  "J          <-'      l  ZT      •  .*•'  yir    '    U 

1  hus  -—  =     ;  hence  E  =     C=-  m. 

64  44 

#      6    ,  /x    '  8  /*     8 

_  =  -;  hence  6r  =  -Cr=  -w. 

<?      6   ,  ,      5         '">    o          5 

—  —  —  ;  hence  A  =  -  c  =  -  .  2m  —  -  m. 
A      5  66  3 

Pursuing  the  same  method  throughout,  the  following 
numbers  are  found  to  represent  the  relative  vibration- 
frequencies  of  the  several  notes  of  the  gamut  : 

Vibration  No.  .  .  64  72  80  85|  96  106 
Name  of  Note  .  .  G  D  E  F  G  A 
Vibration-rate  .  .  m  m  m  w 


Intervals     ....         |          )£        if         f    . 

If  the  fractions  representing  the  vibration-frequencies 
are  reduced  to  a  common  denominator,  the  numerators  may 
then  be  taken  to  denote  the  relative  vibration-frequencies 
of  the  eight  notes.  They  are 

24,  2T,  30,  32,  36,  40,  45,  48. 

1  The  letters  denoting1  the  notes  arc  here  made  to  stand  for  the  vibration- 
frequencies  also. 


PHYSICAL    THEORY  OP  MUSIC.  179 

If  the  above  intervals  between  the  successive  notes  of 
the  scale  are  examined,  it  will  be  seen  that  there  are  only 
three  different  ones  throughout  the  perfect  diatonic  scale. 

The  intervals  —  and  .     are  called  whole   tones,  and  —  a 
8  9  To 

half  tone,  or  a  limma.     The  difference  between  the  two 

81 
whole  tones  is  —  ,  a  comma.     This  is  of  course  the  ratio 

80 

9          10 

between  —  and  •— .     If,  for  example,  the  interval  from  m  to 
8  d 

10  9 

n  is  -—  and  from  m  to  r,  -  ;  then  the  interval  from  n  to  r 
9  8 

.     81 
18   SO" 

The  intervals  between  C  and  each  of  the  other  notes 
in  succession  are  called  a  second,  a  major  third,  a  fourth,  a 
fifth,  a  major  sixth,  a  seventh,  and  an  octave.  The  minor 
intervals  are  counted  backward  from  the  last  note  of  the 
scale.  Thus  the  interval  between  A  and  c  is  a  minor  third. 


14O.  Minor  Chords  and  Transition.  —  The  interpo- 
lated notes,  additional  to  the  eight  of  the  diatonic  scale,  are 
rendered  necessary  in  order  to  provide  for  minor  chords 
and  to  be  able  to  pass  from  a  scale  in  one  key  to  that  in 
another,  a  process  which  is  called  transition.  The  middle 
note  of  a  minor  triad  is  lower  than  that  of  a  major  by  an 

25 

interval  of  — .  Three  interpolated  notes  become  neces- 
sary in  the  key  of  (7,  viz.,  three  notes  below  E,  B,  A,  by 
the  above  interval. 

But  if  we  suppose  the  gamut  to  begin  with  6r,  then  the 
seven  other  notes  must  follow  with  the  same  succession  of 


180  SOUND. 

intervals  as  in  the  key  of  C ;  that  is,  —  ,  -—,   — ,  etc.     Or 

8       9      lo 

in  other  words  for  the  key  of   6r,  the  three  sets  of  major 
triads  are 


•:  B:  d  ) 

:  F:  A   I::  4:  5:  6. 

:  E:    a  ) 


Comparing  these  with  the  three  triads  for  the  key  of  C 
it  will  be  seen  that  two  of  them  are  identical,  while  the 
third  contains  two  notes,  F  and  A,  differing  from  the  scale 
in  the  key  of  C.  A  numerical  comparison  of  the  two 
scales  shows  the  exact  difference. 

Key  of  C. 

<*,   d',  «',   /,   #',   a',   I',   c",  d",  e",  /",  g», 
256,  288,  320,  341J,  384,  426§,  480,  512,  576,  640,  682f,  768 

Key  of  a. 
256,  288,  320,  360,    384,  432,  480,  512,  576,  640,  720,   768 

QA 

The  interval  between  the   a's  of  the  two  scales  is  — , 

81 

while  the  interval  between  the/s  is  much  larger.  One 
other  new  note  besides  these  two  is  necessary  to  provide 
for  minor  triads.  But  other  keys  are  employed  also,  some 
introducing  a  still  larger  number  of  extra  notes ;  so  that, 
with  all  the  naturals  as  key-notes,  the  scale  would  coin- 
prise  at  least  72  notes  to  the  octave. 

141.  Tempered  Scales  (D.,  390;  Bl.,  137).  — Every 
transition  from  one  key  to  another  more  remote  from  0 
multiplies  the  demand  for  new  tones.  The  number  of 
notes  required  to  provide  for  scales  in  all  keys  is  far  in 


PHYSICAL    THEORY  OF  MUSIC.  •  181 

excess  of  possible  provision  in  an  instrument  with  fixed 
keys  like  the  piano.  Hence  some  system  of  accommoda- 
tion must  be  adopted  by  which  the  number  of  notes  shall 
be  much  reduced  by  changing  the  values  of  the  intervals. 
Such  a  modification  of  the  notes  is  called  tempering. 
Every  system  of  tempering  changes  slightly  the  pitch  of 
each  note,  so  as  to  bring  together  into  one  all  the  inter- 
polated notes  falling  between  any  two  adjacent  ones  of  the 
diatonic  scale.  The  intervals  from  E  to  F  and  from  B  to 
0  being  already  semitones,  no  others  are  interpolated 
there.  The  extra  notes,  therefore,  occur  in  groups  of 
threes  and  twos,  represented  by  the  black  keys  on  the 
piano,  making  thirteen  notes  in  the  scale,  with  jbwelve 
intervals. 

The  system  of  temperament  most  commonly  applied  to 
the  organ  and  piano  is  known  as  the  system  of  equal  tem- 
perament introduced  by  Bach.  It  makes  all  intervals  from 
note  to  note  equal,  and  interpolates  only  one  note  in  each 
whole  tone  of  the  diatonic  scale.  Each  interval  of  a  half 
tone  equals  12\/2  or  1.05946.  The  result  differs  widely 
from  pure  intonation.  On  a  pianoforte  the  thirds,  three 
of  which  are  forced  to  make  an  octave,  are  too  sharp, 
though  their  sharpness  adds  somewhat  to  the  brilliancy  of 
the  music. 

The  difference  between  the  eight  notes  in  the  natural 
scale  of  0  and  the  equally-tempered  scale  of  0  appears 
from  the  following  table  : 

CD  E  F  G  A  B  c 

Natural  .  .  1000  1125  1250  1333.33  1500  1666.66  1875  2000 
Tempered  .  1000  1122.46  1259.92  1334.84  1498.31  1681.79  1887.75  2000 

The  above  numbers  represent  only  relative  vibration- 
frequencies. 


182  SOUND. 

"  Mi\sic  founded  on  the  tempered  scale  must  be  con- 
sidered as  imperfect  music,  and  far  below  our  musical 
sensibility  and  aspirations.  That  it  is  endured,  and  even 
thought  beautiful,  only  shows  that  our  ears  have  been  sys- 
tematically falsified  from  infancy."  1 

It  is  an  incorrect  scale,  "  born  of  transition  in  order  to 
avoid  the  practical  difficulties  of  musical  execution."^ 

142.  Laws  of  the  Transverse  Vibration  of  Strings 
(D.,  400;  A.  and  B.,  375;  V.,  II,  166).  —When  a  dis- 
turbance is  produced  at  any  point  of  a  stretched  string  it 
runs  in  both  directions  to  the  fixed  ends  from  which  it  is 
reflected,  and  passing  back  on  the  opposite  side  is  again 
reflected,  and  finally  arrives  at  the  starting  point.  The 
string  has  then  returned  to  its  initial  condition  of  disturb- 
ance, it  has  executed  one  complete  vibration,  and  each 
half  of  the  pulse  has  traversed  the  length  of  the  string 
twice.  But  the  wave-length  along  the  string  is  the  dis- 
tance travelled  in  the  period  of  one  complete  vibration. 
If  then  I  is  the  length  of  the  string 

X  =  2I, 

or  the  wave-length  for  the  fundamental  tone  is  twice  the 
length  of  the  string.  This  wave-length  has  no  relation  to 
the  wave-length  of  sound  in  air. 

Suppose  a  long,  slender,  and  perfectly  flexible  string, 
without  elasticity  properly  speaking,  to  be  strongly 
stretched  and  to  be  drawn  aside  slightly  from  its  initial 
position  of  rest.  Then  the  force  tending  to  restore  it  to 
its  position  of  equilibrium  is  the  component  of  the  ten- 
sion resolved  in  a  direction  at  right  angles  to  the  length  of 
the  string.  The  displacement  being  small,  this  force  of 

1  Blaserna's  Theory  of  Sound,  p.  140. 


PHYSICAL    THEORY  OF  MUSIC.  183 

restitution  varies  directly  as   the  displacement,  and  the 
motion  is  simple  harmonic. 

The  force  of  restitution  therefore  takes  the  place  of 
elasticity  in  the  formula  for  the  transmission  of  longitudi- 
nal vibrations  (117).  We  may  then  write 


iii  which  k  is  the  tension  in  dynes  per  square  centimetre 
of  cross-sectional  area  of  the  string,  and  d  is  the  density. 

If  Tis  the  tension  in  grammes,  then  k=  ~—2 ,  r  being  the 

radius    of   the    cylindrical   string.     The    formula   for  the 
velocity  of  the  pulse  along  the  string  then  becomes 


We  may  now  put  t  for  the  tension  in  dynes  Tg,  and  m 
for  Trr2^,  the  mass  per  unit  length  of  the  string.     Then 


But  the  vibration    rate  n  equals  — ,    and   X   equals  21. 

A, 

Hence,  substituting, 

1      /T 

n  =  —  \  /— . 

2Z  Vm 

The  number  of  vibrations  per  second  of  such  a  stretched 
string,  for  its  fundamental  or  gravest  tone,  is 

1.  Inversely  proportional  to  its  length. 

2.  Directly  proportional  to  the  square  root  of  the  ten- 
sion. 

3.  Inversely  proportional  to  the  square  root  of  its  mass 
per  unit  length. 

These    theoretical   laws   are    found    to  be  very  exactly 


184  SOUND.      . 

true  for  long,  flexible  cords,  strongly  stretched,  and  par- 
ticularly if  they  are  not  metallic.  But  if  the  cords  are 
short,  thick,  and  lightly  stretched,  the  number  of  vibra- 
tions is  always  higher  than  the  theoretical  number,  and 
it  is  higher  the  greater  the  rigidity  of  the  cord  (Violle,  II, 
188).  This  rigidity  acts  in  effect  somewhat  like  another 
tension  added  to  the  stretching  force  T,  although  the 
assimilation  of  the  rigidity  to  a  constant  tension  is  not 
entirely  exact. 

The  mathematical  theory  for  the  establishment  of  the 
preceding  formulas  assumes : 

1.  That  the  transverse  dimensions  of  the  cord  shall  be 
so  small  that  it  can  be  regarded  as  a  simple,  absolutely 
flexible  thread. 

2.  That  the  cord  is  sufficiently  stretched  and  is  only  so 
slightly  deformed  that  the  variable  forces  of  elasticity  re- 
sulting from  these  deformations  may  be  completely  neg- 
lected relative  to  the  permanent  tension  T. 

143.  The  Vibration  of  Strings  in  Segments  (T.,  93  ; 
V.,  II,  169;  Z.,  154).  —  A  string  is  capable  of  vibrating 
not  only  as  a  whole,  but  also  in  equal  segments ;  and  the 
number  of  such  segments,  when  it  is  made  to  vibrate 
in  a  single  mode,  depends  upon  the  relation  between 
the  periodic  time  of  the  disturbances  applied  to  it  and 
the  speed  with  which  these  disturbances  travel  along  the 
string. 

Take  a  soft,  thick,  twisted  cotton  cord,  from  five  to  ten 
metres  long,  and  fix  one  end  to  a  firm  support.  Holding 
the  free  end  in .  the  hand,  move  the  hand  gently  up 
and  down  to  find  the  natural  period  of  oscillation  of 
the  cord  as  a  whole,  or  in  one  segment.  When  this  is 
found  a  series  of  slight  impulses,  so  timed  as  to  aid  the 


PHYSICAL    THEORY  OF  MUSIC.  185 

oscillations  of  the  cord,  will  cause  it  to  swing  through  a 
wide  amplitude. 

Next,  apply  the  transverse  impulses  twice  as  often, 
keeping  the  cord  stretched  with  the  same  tension.  It  will 
now  divide  into  two  equal  segments.  If  the  motion  of  the 
hand  is  in  a  circle,  the  two  segments  of  the  cord  will  be 
large  spindles  with  almost  no  motion  at  the  middle  point. 
Then  let  the  impulses  be  three  times  as  fast  as  at  first ;  the 
cord  will  divide  into  three  vibrating  segments  and  will 
have  the  appearance  of  Fig.  81.  The  two  ends  and  the 


Fig.  81. 

points,  jV,  JV,  are  called  nodes.  They  are  the  points  of  least 
motion.  The  intermediate  points,  V,  F",  F",  are  called 
antinodes.  Two  points  on  opposite  sides  of  a  node  are 
always  moving  in  opposite  directions.  If  the  motion  of 
every  point  of  the  string  is  circular  then,  while  two  points 
on  opposite  sides  of  a  node  are  moving  in  the  same  direc- 
tion around  their  respective  circles,  one  is  moving  in  one 
direction  in  space  on  one  side  of  a  circle  while  the  other  is 
moving  in  the  other  direction  on  the  other  side  of  its 
circle.  In  other  words,  their  motions  differ  in  phase  by 
half  a  period. 

By  increasing  the  frequency  of  the  movements  of  the 
hand  the  cord  may  be  made  to  divide  into  four,  five,  six, 
or  even  more  segments,  according  to  the  dexterity  of  the 
experimenter. 

When  a  cord  vibrates  in  this  way,  with  fixed  nodes,  it 
illustrates  what  is  known  as  stationary  waves.  Stationary 
waves  result  from  the  superposition  of  two  wave  systems, 


186  SOUND. 

one  direct  and  the  other  reflected.  The  relation  between 
the  speed  of  transmission  of  the  wave  along  the  cord  and 
the  number  of  vibrations  is  easily  found. 

Let  a  transverse  disturbance  be  started  at  one  end ;  it 
runs  along  the  cord  and  is  reflected  at  the  other  end  with 
a  change  of  sign  of  the  motion,  a  protuberance  being 
transformed  into  a  depression.  On  arriving  again  at  the 
origin  or  free  end  it  is  again  reflected  with  a  change  of 
sign.  If  now  this  pulse,  which  has  been  twice  reflected, 
agrees  in  phase  with  another  pulse  just  starting  from  the 
origin,  then  their  motions  will  be  added  together  ;  and  in 
this  way  a  periodic  movement  applied  at  one  end  is  rapidly 
amplified,  the  wave  twice  reflected  being  identified  with 
the  direct  wave,  if  the  period  of  the  double  passage  of  the 
wave  along  the  cord  is  a  whole  number  p  of  periods  of 
vibration.  Let  v  be  the  speed  of  transmission  of  the  trans- 
verse motion  along  the  cord,  I  the  length  of  the  cord 
between  the  two  points  of  reflection  at  the  ends,  and  T  the 
period.  Then  the  condition  of  reenforcement  is 

21        ^ 
-  =  pT. 

v      r 

But  since  w,  the  vibration  number,  is  the  reciprocal  of  T, 

v 

n=pzr 

If  p  is  unity,  the  cord  is  vibrating  in  a  single  seg- 
ment ;  if  p  is  two,  the  cord  divides  into  two  segments  and 
the  period  is  half  as  great  as  before.  If  p  is  three  the  cord 
divides  into  three  segments,  and  each  segment  executes 
three  complete  vibrations  Avhile  the  pulse  travels  over 
twice  the  length  of  the  cord. 

144.  Segmental  Vibration  of  the  Monochord.  —  A 
monochord  is  essentially  a  single  stretched  wire.  It  is 


PHYSICAL    THEORY  OF  MUSIC.  187 

usually  mounted  on  a  box  of  thin  resonant  wood,  with 
lateral  apertures  communicating  with  the  external  air. 
Near  the  ends  of  the  box  are  bridges  and  the  wire  is 
stretched  over  them.  A  sonometer,  as  the  instrument  is 
often  called,  provided  with  three  wires,  is  shown  in  Fig.  82, 


Fig.  82. 


For  certain  purposes  it  is  better  to  take  a  steel  piano  wire, 
about  No.  22  gauge,  and  four  metres  long,  and  stretch  it 
over  two  appropriate  bridges  attached  to  the  top  of  a  long 
table.  With  this  wire  properly  stretched,  the  vibration  in 
segments  may  be  strikingly  illustrated.  A  thin  piece  of 
cork,  about  an  inch  in  diameter  with  a  small  hole  at  the 
centre,  should  slide  readily  along  the  wire.  Provide  little 
riders  of  stiff  paper  bent  double,  some  white  and  some  red. 
Let  the  slip  of  cork  be  placed  one  metre  from  one  end,  and 
let  white  riders  be  placed  at  the  middle  and  at  one  metre 
from  the  other  end,  while  red  ones  are  mounted  on  the 
wire  at  intermediate  points.  Now  touch  the  cork  very 
lightly  and  draw  a  heavy  bow  across  the  short  division  of 
the  wire  — the  metre  length  at  one  end.  If  this  is  deftly 
done,  the  wire  will  sound,  the  red  riders  will*  be  violently 
unhorsed,  while  the  white  ones  will  remain  in  place  on  the 
wire.  The  white  riders  therefore  mark  the  place  of  the 
nodes,  and  the  string  vibrates  in  four  segments,  each  a 
metre  in  length.  If  the  cork  slip  is  placed  at  80  cms.  from 
one  end  and  the  white  riders  are  mounted  at  distances  of 
80  cms.  apart  along  the  wire  so  as  to  mark  the  places  of 
five  equal  divisions,  then,  upon  agitating  the  first  segment 


188  SOUND. 

of  the  wire  by  the  bow  as  before,  the  intermediate  red 
riders  will  be  thrown  off,  while  the  white  ones  will  remain 
sitting.  This  method  of  exhibiting  nodes  and  antinodes 
was  first  employed  apparently  by  Noble  and  Pigott  at 
Oxford  in  1673,  but  the  application  of  it  to  a  monochord 
was  made  by  Sauveur  in  1701. 

In  this  experiment  impulses  of  the  proper  period  are 
obtained  by  the  vibration  of  the  short  segment  of  the  wire, 
which  must  of  course  then  be  an  aliquot  portion  of  the 
whole.  This  segment  furnishes  the  timed  impulses  for  the 
remainder  of  the  wire,  and  the  position  of  the  cork  slip 
must  be  considered  the  origin,  corresponding  with  the  free 
end  of  the  cotton  cord  held  in  the  hand  as  already  de- 
scribed. The  relationship  between  period  and  speed  of 
transmission  obtained  in  the  last  section  must  then  clearly 
hold  true  for  this  case  of  the  monochord. 

The  origin,  marked  by  the  cork,  which  serves  to  produce 
a  node  at  the  point,  is  not  a  place  of  no  motion,  but  of 
minimum  motion ;  the  small  movements  transmitted  across 
it  are  accumulated  and  amplified  in  the  other  segments  by 
the  addition  of  the  direct  waves  to  the  reflected  ones. 

A  stretched  wire  or  string  may  thus  vibrate  in  any  num- 
ber of  equal  segments.  The  number  of  vibrations  executed 
per  second  will  be  proportional  to  the  number  of  segments 
into  which  the  wire  divides.  Thus  the  vibration-frequency 
for  three  segments  will  be  three  times  as  great  as  for  the 
fundamental  of  one  segment;  for  four  segments,  four 
times  as  great,  etc. 

145.  Melde's  Experiments  (V.,  II,  17O;  D.,  4O5  ;  Z., 
157). — A  long  white  silk  cord  is  stretched  horizontally 
between  a  small  fixed  pulley  and  a  vertical  tuning-fork. 
The  plane  of  the  two  branches  of  the  diapason  contains 


PHYSICAL    THEORY  OF  MUSIC.  189 

the  cord,  the  end  of  —hich  is  displaced  longitudinally 
when  the  fork  vibrates.  The  cord  relaxes,  falls  to  its 
lowest  position  with  the  forward  movement  of  the  fork, 
again  rises  to  the  horizontal,  and  then  to  its  highest  posi- 
tion, when  the  fork  is  again  in  its  most  forward  position. 
The  longitudinal  movement  of  the  point  of  attachment 
thus  gives  rise  to  a  transverse  motion  of  the  cord,  with 
a  period  double  that  of  the  fork.  The  entire  cord  will 
oscillate  then  an  octave  below  the  tuning-fork.  For  this 
purpose  the  tension  must  be  carefully  adjusted  by  weights 
in  a  scale  pan  hung  on  the  cord  beyond  the  pulley.  When 
the  exact  tension  required  has  been  found  the  cord  spreads 
out  in  a  pearl-Avhite  spindle,  which  appears  to  be  perfectly 
fixed  and  stable. 

If  now  the  fork  be  turned  on  its  axis  so  that  it  commu- 
nicates transverse  impulses  to  the  cord,  the  conditions  then 
obviously  require  the  fork  and  cord  to  vibrate  in  unison. 
The  cord  will  then  break  up  into  two  segments  separated 
by  a  node.  Each  half  vibrates  twice  as  fast  as  the  entire 
cord,  and  so  keeps  in  unison  with  the  fork.  This  demon- 
strates the  law  of  lengths. 

Next  turn  the  fork  back  into  its  former  position.  By 
reducing  the  weights,  including  the  pan,  to  one-quarter, 
the  cord  again  divides  into  two  segments.  Each  segment 
again  vibrates  at  a  rate  equal  to  an  octave  below  the  fork. 
As  a  whole  string  it  would  vibrate  two  octaves  below ;  but 
by  dividing  into  two  segments  its  rate  remains  the  same  as 
at  first,  while  the  time  required  for  the  pulse  to  travel 
twice  its  length  has  been  doubled  by  reducing  the  ten- 
sion. 

With  the  tension  reduced  to  one-ninth  the  cord  divides 
into  three  segments.  Since  in  each  case  the  segments 
vibrate  an  octave  lower  than  the  fork,  and  since  for  a 


190  SOUND. 

tension  of  one-ninth,  for  example,  the  cord  divides  into 
three  segments,  or  the  vibration-frequency  is  reduced  to 
one-third,  the  law  of  tensions  is  thus  verified. 

PROBLEMS. 

1.  Suppose  a  string,  vibrated  by  a  tuning-fork,  is  stretched  with 
a  weight  of  270  gms.  and  divides  into  four  segments.     What  must 
be  the  weight  to  cause  it  to  divide  into  three  segments  with  the  same 
fork? 

2.  A  cord  attached  to  a  fork  with  its  plane  of  vibration  in  the 
direction  of  the  string  divides  into  two  segments  when  stretched 
with  270    gms.     With  the  plane  of  vibration  of  the  fork  at  right 
angles  to  the  cord,  what  weight  must  be  applied  to  cause  it  to  divide 
into  three  segments  ? 

3.  A  cord  vibrates  synchronously  with  the  attached  fork  by  divid- 
ing into  three  segments.     If  it  be  replaced  by  a  similar  one  of  the 
same  length  and  four  times  the  sectional  area,  what  relative  weight 
will  be  required  to  cause  it  to  divide  into  four  segments  ? 

146.  Overtones.  --  When  a  stretched  string  or  wire  is 
made  to  vibrate  it  not  only  gives  its  fundamental  tone,  but 
it  divides  at  the  same  time  into  one  or  more  sets  of  equal 
segments,  which  produce  higher  tones  than  the  funda- 
mental ;  so  that  usually  there  are  several  series  of  vibra- 
tions superposed  upon  the  fundamental  one.  The  tones 
of  higher  pitch  associated  with  the  fundamental  are  known 
by  the  general  name  of  overtones  or  upper  partials. 

In  the  case  of  strings  the  division  will  be  into  two, 
three,  four,  five,  etc.,  equal  segments,  with  vibration- 
frequencies  two,  three,  four,  five,  etc.,  times  that  of  the 
fundamental  tone.  The  intervals  between  the  funda- 
mental and  the  overtones  are  therefore  an  octave,  an 
octave  plus  a  fifth,  or  a  twelfth,  a  double  octave,  two 
octaves  plus  a  major  third,  two  octaves  plus  a  fifth,  etc. 
If,  for  example,  the  fundamental  is  (7,  the  first  overtone 


PHYSICAL    THEORY   OF  MUSIC.  191 

is  cy,  the  second  #',  the  third  c",  the  fourth  e",  and  the  fifth 
</",  while  the  sixth  overtone  having  a  frequency  of  vibra- 
tion seven  times  the  fundamental,  is  not  represented  by 
any  tone  in  the  diatonic  scale  or  gamut.  The  eighth 
overtone,  with  nine  times  the  frequency  of  the  funda- 
mental, is  d'",  which  produces  a  discord  with  the  fundamen- 
tal tone.  The  formation  of  these  particular  overtones  is 
prevented  on  the  pianoforte  by  having  the  hammer  strike 
the  wire  at  a  distance  of  a  little  less  than  one-seventh  the 
length  of  the  string  from  one  end.  Since  the  point  struck 
must  be  an  antinode  for  every  system  of  subdivision,  all 
modes  of  segmental  vibration  requiring  a  node  at  the  point 
struck  are  thereby  eliminated. 

147.  Distinction  between  Partial  Tones  and  Har- 
monics.1—  When  the  vibrating  parts  of  a  musical  instru- 
ment which  is  producing  composite  tones,  such  as  the 
strings  of  a  piano  or  a  violin,  or  the  column  of  air  in  an 
organ  pipe,  divide  into  several  series  of  segments  at  the 
same  instant,  all  the  tones  produced  by  these  segmental 
vibrations  are  called  partial  tones.  But  the  vibrating  parts 
of  many  musical  instruments  may  execute  a  variety  of 
very  complex  and  imperfectly  pendular  motions,  which  are 
not  made  up  by  the  superposition  of  several  series  of  equal 
subdivisions.  All  such  vibrations,  however,  in  order  to 
produce  musical  sounds,  must  have  the  characteristic  of 
periodicity  ;  that  is,  they  must  repeat  themselves  over  and 
over  in  certain  definite  and  equal  intervals  of  time.  Such 
complex  periodic  motions  are  subject  to  the  following 
law  of  Fourier :  Every  periodic  motion  whatsoever  may 
always  be  considered  as  the  resultant  of  the  superposition  of  a 

1  Koenig's  Quelques  Experiences  d'Acoustique,  218;  Wiedemann's  Annalen, 
1881. 


192  SOUND. 

definite  number  of  pendular  vibratory  motions,  or  is  ahvays 
resolvable  into  a  definite  number  of  commensurate  simple  har- 
monic motions.  The  frequencies  of  these  simple  harmonic 
components  of  the  complex  periodic  motion  are  all  exact 
multiples  of  the  fundamental ;  that  is,  if  the  period  of  the 
fundamental  tone  is  T,  then  the  periods  of  the  overtones 

T     T    T    T 

must  all  fall  in  the  series  — ,    — ,    — ,    -,    etc.,    or  their 

2345 

vibration-frequencies  are  2,  3,  4,  5,  etc.,  times  that  of  the 
gravest  tone.  Now  the  component  tones  due  to  these  higher 
frequencies,  which  are  rigorously  exact  multiples  of  the 
fundamental,  Koenig  calls  harmonics.  It  is  true  that  the 
word  harmonic  is  often  applied  to  those  partial  tones  which 
harmonize  with  the  fundamental,  but  that  is  not  the 
meaning  attached  to  the  term  here.  Koenig  says  :  "  Among 
the  sounds  into  which  the  sonorous  mass,  which  emanates 
from  a  vibrating  body,  may  be  resolved,  we  may  distinguish 
harmonics  and  partial  tones.  These  last  have  their  origin 
when  the  body  in  question  executes  simultaneously  several 
modes  of  vibration  which  it  can  adopt  separately,  as  in  the 
case  of  the  string;  while  the  harmonics  are  due  to  the 
resolution  into  simple  pendular  motions  of  the  imperfectly 
pendular  oscillations  of  the  sounding  body  executing  a 
single  mode  of  vibration."  Partial  tones  are  therefore  due 
to  the  actual  subdivision  of  the  sonorous  body  into  vibrat- 
ing segments;  harmonics,  on  the  contrary,  are  the  com- 
mensurate components  of  the  motion  when  the  body 
vibrates  periodically,  but  by  only  one  mode,  arid  that  a 
complex  one.  Harmonics  have  m  frequencies  which  are 
necessarily  exact  multiples  of  the  fundamental;  the  fre- 
quencies of  partial  tones  may  or  may  not  be  exact  mul- 
tiples. Even  if  they  are  exact  multiples,  still  they 
originate,  each  in  a  corresponding  subdivision  of  the 


PHYSICAL    THEORY   OF  MUSIC.  193 

sonorous  body  or  source  of  sound,  while  the  harmonics 
have  no  such  physical  foundation.  They  are  only  the 
components  into  which  mathematical  analysis  shows  that 
the  motion  is  resolvable. 

Harmonics  are  always  due  to  frequencies  represented  by 
the  series  of  exact  whole  numbers,  while  the  frequencies 
of  partial  tones  approach  only  more  or  less  nearly  to  their 
theoretical  values.  Two  diapasons,  whose  fundamentals 
are  very  exactly  in  unison,  may  give  partial  tones  of  the 
same  order  which  produce  loud  beats,  and  which  are  there- 
fore not  in  unison.  If  the  overtone  of  one  of  the  diapasons 
is  an  exact  multiple  of  its  fundamental,  that  of  the  other 
cannot  be. 

Partial  tones  then  are  not  rigorously  exact  multiples  of 
the  fundamental  in  respect  to  their  frequencies  of  vibra- 
tion. The  frequencies  of  vibration  of  all  the  segments 
into  which  a  string,  for  example,  divides  are  not  neces- 
sarily or  exactly  equal  to  each  other.  The  variation  from 
such  equality  may  be  due  to  variation  in  cross-section  of 
the  wire  or  string,  to  variation  in  density  or  hardness,  in 
the  physical  or  chemical  state  of  the  carbon  present  in  dif- 
ferent parts  of  the  wire,  etc.  The  wire  presenting  the 
greatest  uniformity  in  all  respects  throughout  its  length 
will  give  the  best  tone  by  producing  partial  tones  which 
are  as  nearly  as  possible  multiples  of  the  fundamental. 

148.  The  Transverse  Vibration  of  Rods  (V.,  II,  195  ; 
B.,  235;  D.,  4O7).  —  Rods  vibrating  transversely  may 
have  a  circular,  square,  or  rectangular  section.  A  rod  of 
circular  section  vibrates  transversely  in  all  directions 
without  difference.  One  with  a  rectangular  section  vi- 
brates with  larger  amplitude  in  a  plane  at  right  angles 
to  the  broad  face  than  in  the  plane  parallel  to  this. face. 


194  SOUND. 

In  the  transverse  vibration  of  rods,  unlike  that  of 
strings,  the  force  of  restitution  is  the  elasticity  of  flexure. 
The  theory  is  complex,  but  the  number  of  vibrations  per 
second  is  given  by  the  equation 

n  =  G~i- 

C  is  a  constant  which  depends  upon  the  manner  of  sup- 
porting the  rod.  If  the  rod  is  free  or  clamped  at  both 
ends  0  is  1.78  ;  if  free  at  one  end  only  it  is  0.28.  For  the 
other  terms,  t  is  the  thickness  of  the  rod,  I  its  length,  e  its 
coefficient  or  modulus  of  elasticity,  and  d  its  density. 

For  rods  of  the  same  thickness  the  frequency  of  vibra- 
tion is  inversely  as  the  square  of  the  length ;  but  if  the 
thickness  and  the  length  vary  in  the  same  ratio  the  fre- 
quency is  inversely  as  the  length.  A  tuning-fork  five  cms. 
long  gives  a  note  an  octave  above  one  ten  cms.  long,  pro- 
vided the  two  forks  have  the  same  relative  dimensions. 
The  same  rule  applies  to  reeds. 

The  vibration-frequency  of  a  rod  is  independent  of  its 
width,  but  is  directly  proportional  to  its  thickness.  Hence 
if  two  rods  or  bars  of  the  same  material  have  the  same 
length,  while  one  is  twice  as  thick  as  the  other,  the  thick 
one  will  vibrate  in  half  the  period  or  with  twice  the  fre- 
quency, whatever  may  be  their  relative  widths. 

The  partial  tones  of  rods  rise  much  more  rapidly  than 
those  of  strings.  For  a  rod  fixed  at  one  end  and  free 
at  the  other  Chladni  found  the  following  relative  fre- 
quencies : 

1         6.25        17.5        34.25         56.5         84 
or  (1-2)2       32  52  72  92          112 

Examples  of  the  use  of  transversely  vibrating  rods  in 
musical  instruments  are  the  reeds  of  an  accordion  or  liar- 


PHYSICAL    THEORY  OF  MUSIC. 


195 


monium,  the  tongue  of  a  jew's-harp,  or  of  a  music  box, 
the  reeds  of  reed-pipes  in  organs,  the  claque-bois  or  xylo- 
phone, and  the  tuning-fork. 

The  xylophone  is  a  primitive  instrument  with  rods  free 
at  both  ends.  It  consists  of  a  series  of  small  wood  prisms 
of  convenient  length  and  thickness,  supported  by  strings 
at  the  nodes,  which  are  about  one-quarter  of  the  length 
from  each  end.  The  prisms  are  adjusted  to  give  the  notes 
of  the  scale.  They  are  played  by  striking  them  in  the 
middle  with  a  light  hammer  having  a  soft  elastic  face. 

149.  The  Diapason  or  Tuning-Fork.  -  -  The  tuning- 
fork  is  one  of  the  most  important  applications  of  vibrating 
rods  free  at  both  ends.  A  straight  elastic  bar  when 
sounding  its  lowest 
note  has  two  nodes 
each  at  a  distance 
from  the  end  of  about 
one-fourth  the  dis- 
tance between  them. 
As  this  rod  is  grad- 
ually bent  into  the 
form  of  a  tuning-fork* 
(Fig.  83),  the  nodes 
approach  each  other; 
and  when  the  fork 
is  provided  with  a 
stem  the  nodes  are  near  the  bottom  of  each  branch.  The 
two  branches  then  vibrate  in  unison,  each  comporting 
itself  sensibly  like  a  rod  free  at  one  end  and  fixed  at  the 
other.  The  stem  or  base  of  the  fork  has  a  slight  up  and 
down  motion,  which  is  transmitted  to  the  resonant  box  on 
which  it  is  mounted. 


Fig.  83. 


196  SOUND. 

The  vibration-frequency  of  the  fork  is  independent  of 
the  breadth  of  the  branches,  but  is  directly  proportional 
to  their  thickness  measured  in  the  plane  of  vibration,  and 
inversely  proportional  to  the  square  of  their  length, 

T^t 

or  n  =  K  - , 

where  Kis  a  constant  which  equals  for  steel  about  82,000, 
the  unit  of  length  being  the  centimetre. 

The  partial  tones  succeed  one  another  according  to  the 
law  of  Chladni  already  given,  viz., 

1,  6.25,  7.5,  34.25,  56.5. 

The  first  partial  tone  of  a  fork,  giving  for  its  fundamental 
256  vibrations  per  second,  will  be  256  x  6.25  =  1600.  The 
first  overtone  corresponds  to  the  presence  of  two  nodes  on 
each  branch,  the  second  overtone  to  three,  etc.  The  ratio 
of  the  first  overtone  to  the  fundamental  varies  somewhat 
on  different  diapasons.  Tyndall  found  these  values  to  be 
comprised  between  5.8  and  6.6. 

The  diapason  is  the  true  standard  of  the  musical  scale. 
That  the  pitch  of  the  sound  of  a  diapason  may  be  abso- 
lutely definite,  it  is  necessary  that  the  amplitude  of  the 
oscillations  be  very  small,  not  exceeding  TJ^  of  the  length 
of  the  branches,  and  the  temperature  must  remain  con- 
stant. Koenig  adjusts  all  his  diapasons  at  20°  C.  Ac- 
cording to  Koenig  the  frequency  of  a  steel  diapason 
diminishes  -g^V  ¥  when  the  temperature  rises  1°  C. 

The  isochronism  of  the  small  oscillations  of  a  fork,  at  a 
constant  temperature,  furnishes  a  very  exact  chronogruj  h 
for  recording  small  intervals  of  time.  For  this  purpose  the 
vibration  must  be  maintained  with  the  same  amplitude  as 
nearly  as  possible. 

This  is  effected  readily  by  means  of  electricity  and  an 
electro-magnet. 


PHYSICAL    THEORY  OF  MUSIC. 


197 


15O.  The  Transverse  Vibration  of  Plates  (K,,  32 ; 
V.,  II,  229;  B.,  237;  Tyn.,  139). — If  a  square  or  a 
round  plate  of  elastic  material,  such  as  glass  or  brass,  be 
clamped  at  the  centre  in 
a  horizontal  position, 
and  sand  be  scattered 
upon  it,  this  sand  will 
gather  along  certain 
definite  nodal  lines  (Fig. 
84)  when  the  plate  is 
made  to  emit  sound  by 
bowing  on  the  edge. 
These  sound-figures 
were  first  obtained  by 
Chladni,  and  are  known 
as  Chladni's  figures. 
The  explanation  of  them 
involves  many  difficulties.  For  certain  of  the  simpler 
figures,  the  explanation  given  by  Wheatstone  will  suffice 
here. 

Consider  a  long  narrow  plate,  free  at  both  ends,  and 
vibrating  transversely  so  as  to  give  its  fundamental  tone. 
It  has  then  two  nodal  lines  running  across  it  at  a  distance 
of  0.224  of  its  length  from  the  ends.  The  width  of  this 
plate,  which  is  essentially  a  rod  or  bar,  does  not  affect  its 
frequency  within  wide  limits.  Let  us  therefore  assume  a 
width  equal  to  the  length.  We  have  then  a  square  plate 
with  a  system  of  nodal  lines  parallel  to  two  opposite  sides. 
But  unless  the  plate  is  clamped  entirely  across  its  surface, 
a  disturbance  at  any  point  is  as  likely  to  start  vibrations 
with  nodal  lines  parallel  to  one  pair  of  sides  as  to  the 
other ;  we  may  therefore  suppose  that  there  are  two  sys- 
tems of  vibrations  superposed  on  the  same  plate  crossing 


198 


SOUND. 


each  other  at  a  right  angle.  We  have  then  a  new  mode  of 
vibratory  motion  characterized  by  a  system  of  nodal  lines 
passing  through  the  nodal  points  common  to  the  two 
systems,  and  through  those  points  where  the  motion  is  zero 
because  of  the  reciprocal  action  of  the  two  superposed 
motions.  This  method  offers  certain  advantages,  in  spite 
of  its  defects,  for  a  first  or  approximate  explanation  of  the 
phenomena. 

Let  the  two  pairs  of  dotted  lines  in  Fig.  85  represent  the 
nodal  lines  for  the  two  systems  of  superposed  vibrations. 
These  may  coexist  in  two  ways.  If  the  two  systems  are 
superposed  as  shown  in  Fig.  85,  where  the  sign  -f-  means  a 

motion  upward  and  —  a 
motion  downward,  then 
the  nodal  lines  of  the 
resulting  system  will  be 
those  of  3,  Fig.  86.  For 
the  four  points  of  inter- 
section of  the  two  pairs 
of  rectangular  nodal 
lines  will  be  points  on  the  resulting  nodes;  the  middle 
point  of  each  side  will  also  be  on  the  new  nodal  lines, 
because  the  motions  of  the  two  systems  are  then  in  oppo- 
site directions.  Con- 
necting these  points  to- 
gether the  result  is  the 
square  of  3,  Fig.  86. 

If,  however,  the  mo- 
tions of  one  of  the  su- 
perposed systems  of 
Fig.  85  changes  sign, 
then  the  resultant  lines  of  minimum  motion  will  be  the 
two  diagonals  of  4,  Fig.  86,  since  then  the  corners  of  the 


Fig.  85. 


Fig.  86. 


PHYSICAL    THEORY  OF  MUSIC.  199 

square  will  be  affected  by  motions  of  opposite  sign  in 
the  two  component  systems.  The  former  figure,  with  the 
angles  rounded  off,  has  been  obtained  by  clamping  the 
plate  near  the  middle  of  one  edge,  and  bowing  it  at  one 
of  the  angles. 

The  other  figure  is  readily  produced  by  clamping  the 
plate  at  the  centre,  holding  the  finger  against  one  corner,  and 
bowing  slowly  at  the  middle  point  of  an  adjacent  edge. 

The  two  systems  of  superposed  waves  at  right  angles  may 
be  shown  by  clamping  a  glass  plate  at  the  middle,  and 
after  carefully  levelling,  spreading  over  it  a  thin  layer  of 
water.  When  the  plate  is  vibrated  the  surface  of  the 
water  is  agitated  with  stationary  waves  in  the  form  of  a 
square  check,  showing  plainly  the  coexistence  of  the  two 
rectangular  systems  of  motion.  Even  a  film  of  soapy 
water  in  a  square  opening  will  show  similar  stationary 
figures  when  thrown  into  vibration  by  any  appropriate 
sound.  A  triangular  opening,  covered  by  a  film,  gives  rise 
to  three  sets  of  plain  waves,  which,  together  with  their 
reflected  systems,  produce  stationary  waves  of  a  hexagonal 
pattern. 

When  a  round  plate  is  clamped  at  its  centre,  its  funda- 
mental tone  is  produced  by  a  division  into  four  equal  seg- 
ments by  two  diameters;  the  first  overtone  is  due  to  a 
division  into  six  segments  by  three  diameters,  the  second 
by  eight  segments  and  four  diameters,  etc.  Adjacent 
segments,  like  those  of  strings,  are  always  in  opposite 
phases  of  motion. 

With  rectangular  plates  whose  sides  have  such  rela- 
tive dimensions  that  the  two  wave  systems  have  frequen- 
cies represented  by  some  simple  ratios,  Wheatstone  and 
Koenig  have  obtained  figures  approximating  very  closely 
to  the  theoretical  Lissajous  curves. 


200  SOUND. 

151.  Resonance  (V.,  II,  279;  Z.,  266;  Bl.,  51).- 
When  vibrations  come  to  an  elastic  body  in  accord  with 
those  which  it  can  itself  execute,  it  is  set  vibrating  as  a 
whole ;  and,  under  the  repeated  action  of  the  synchronous 
impulses,  it  may  oscillate  in  complete  unison  with  the  ex- 
ternal vibrations.  This  is  resonance.  Resonance  depends 
upon  the  cumulative  effect  of  small  disturbances  when 
applied  to  a  body  in  such  a  way  as  to  synchronize  with  its 
own  motions.  One  string  thus  takes  up  the  vibrations  of 
another  which  lias  the  same  vibration-rate.  When  two 
heavy  pendulums  are  hung  on  the  same  stand  and  adjusted 
to  swing  in  exactly  the  same  period,  the  motion  of  one  of 
them  will  be  communicated  to  the  other.  One  drags 
slightly  behind  the  other  and  absorbs  its  energy  till  the 
first  one  comes  nearly  to  rest.  The  process  is  then  reversed. 
An  organ  will  often  throw  windows  into  loud  vibration, 
producing  a  rattle.  If  two  tuning-forks,  mounted  on 
reenforcing  cases,  are  adjusted  to  exact  unison,  the  phenom- 
enon of  resonance  is  easily  demonstrated  at  a  distance  of 
several  metres  between  them.  When  one  has  been  bowed 
and  is  then  stopped  by  touching  it,  the  other  will  be  found 
to  be  producing  a  very  audible  sound.  The  impulses 
setting  the  second  fork  in  motion  may  even  be  transmitted 
to  a  distance  by  electricity  along  a  wire  instead  of  through 
the  air. 

Every  elastic  body  has  its  own  rate  of  vibration,  depend- 
ing upon  its  coefficient  of  elasticity,  its  density,  and  its 
dimensions.  A  mass  of  confined  or  enclosed  air  has  its 
own  period  of  vibration.  Hold  a  common  A  tuning-fork 
over  the  mouth  of  a  tall  cylindrical  jar,  and  while  the  fork 
vibrates  pour  in  water  slowly.  As  the  air  column  shortens 
the  sound  increases  in  loudness  up  to  a  definite  point, 
beyond  which  the  further  shortening  of  the  column  of  air 


PHYSICAL    THEORY   OF  MUSIC.  201 

in  the  jar  diminishes  the  sound.  If  forks  of  different  pitch 
are  tried,  each  one  will  be  found  to  have  its  own  length  of 
air  column  which  will  reenforce  its  sound.  This  increase 
in  the  volume  of  sound,  due  to  the  synchronous  vibration 
of  another  body,  usually  a  mass  of 
partly  enclosed  air,  is  resonance. 

The  resonators  of  von  Helm- 
holtz  (Fig.  87)  are  very  valuable 
for  researches  in  sound.  They  con- 
sist of  spheres,  provided  with  two 
opposite  tubular  openings  ;  one  is 
short  and  straight,  making  free 
communication  with  the  outer  air; 

the  other  is  small  and  bell-shaped,  so  as  to  be  introduced 
into  the  ear,  where  it  is  closed  by  the  membrane  of  the 
tympanum.  Each  resonator  is  adjusted  in  dimensions  to 
respond  as  a  fundamental  to  some  particular  tone  ;  it  is 
then  practically  responsive  to  only  a  single  tone,  and 
whenever  this  one  is  present  it  declares  itself  in  the  ear 
with  a  prodigious  force.  By  this  means  one  can  distin- 
guish the  presence  of  a  feeble  sound  in  the  midst  of  many 
other  loud  ones.  The  "  sound  of  the  sea,"  heard  in  a  sea 
shell,  is  a  similar  reinforcement  by  selective  absorption  of 
vibrations. 

152.  The  Length  of  Organ  Pipes  and  the  Wave- 
Length  of  their  Fundamental  Tone.  —  If  we  take  several 
glass  or  metal  tubes  of  different  length  and  about  two 
cms.  in  diameter  and  blow  sharply  across  the  edge  at  one 
end,  while  the  other  end  is  closed,  we  shall  find  that  the 
different  tubes  give  sounds  of  different  pitch,  the  longer 
the  tube  the  graver  the  sound  of  its  fundamental  tone. 
The  air  in  each  tube  has  a  definite  rate  of  vibration,  and 


B 


202  SOUND. 

when  by  blowing  across  the  tube  a  flutter  is  produced  at 
one  end,  consisting  of  disturbances  of  various  frequencies 
mingled  together,  the  air  column  of  the  tube  selects  for 
reinforcement  the  disturbance  of  its  own  rate  and  exalts 
that  into  a  musical  sound.  This  reenforcement  is  accom- 
plished by  means  of  resonance. 

For  the  purpose  of  analyzing  the  actions  going  on  in  the 
pipe,  suppose  a  reed  vibrating  at  the  middle  of  a  tube 
closed  at  both  ends  (Fig.  88),  and  that  its  rate  corresponds 

with  that  of  the  en- 
closed  mass  of  air,  or 
that  it  covibrates  with 
it.  Let  the  reed  be 
drawn  aside  to  the  po- 

Fig.  88. 

.  sition  a".     Then  while 

it  moves  from  a"  to  a'  let  the  condensation  in  front  of 
it  run  from  the  reed  to  the  end  A,  and  after  reflection 
back  to  the  reed  again.  Similarly  while  the  reed  moves 
from  a'  to  a"  let  the  condensation  run  from  it  to  B  and 
back  again  to  the  middle  of  the  pipe.  The  reed  has  now 
executed  one  complete  vibration,  the  condensation  has  run 
over  the  length  of  the  pipe  twice,  and  the  two  are  ready 
to  repeat  the  process.  The  condensation  which  has  been 
twice  reflected  is  at  the  middle  of  the  pipe  and  moving 
toward  A.  It  is  thus  ready  to  join  the  second  condensa- 
tion produced  by  the  reed  and  running  toward  A.  The 
motion  of  the  rarefaction  may  be  similarly  traced,  the  con- 
densation running  in  one  direction  while  the  rarefaction 
runs  in  the  other.  Since  the  disturbance  traverses  the 
pipe  twice  during  a  complete  vibration  of  the  reed, 
the  wave-length  of  the  sound  is  twice  the  length  of  this 
pipe. 

If  now  the  pipe  be  cut  in  two  at  the  middle  and  the 


PHYSICAL    THEORY  OF  MUSIC.  203 

reed  vibrate  at  the  open  end,  the  condensation  Xvill  run 
into  the  pipe  and  back  to  the  reed  during  the  forward 
excursion  from  a"  to  a',  and  the  rarefaction  will  then  run 
in  and  back  during  the  return  movement  of  the  reed  from 
a'  to  a".  The  initial  conditions  then  recur.  The  dis- 
turbance traverses  the  pipe  four  times  during  a  vibration 
of  the  reed.  Such  a  tube  corresponds  with  a  closed  organ 
pipe,  which  is  closed  at  one  end  only.  The  stopped 
organ  pipe  is  therefore  one-fourth  the  wave-length  of  its 
fundamental  tone  in  air. 

If  the  reed  be  supposed  to  vibrate  at  one  end  of  a  pipe 
open  at  both  ends  (Fig.  89),  then  while  the  reed  moves  from 
a'1  to  a'  the  conden- 
sation runs  the  en- 
tire   length    of    the 
pipe    to   A    and   is 

there  reflected  as  a  Fig.  39. 

rarefaction ;  that  is, 

the  condensation  changes  sign  while  the  motion  of  the 
air  particles,  by  which  the  rarefaction  is  propagated  back- 
ward, has  not  changed  sign.  It  continues  in  the  direction 
from  B  toward  A.  While  now  the  rarefaction  as  a  re- 
flected wave  runs  from  A  toward  B,  the  reed  by  its  motion 
from  a'  to  a  sends  another  rarefaction  into  the  tube. at  the 
end  B.  The  two  meet  at  the  middle  of  the  tube,  produc- 
ing a  node,  and  are  reflected  from  this  to  the  two  open 
ends  of  the  pipe  ;  so  that  when  the  reed  reaches  a"  the 
rarefaction  has  returned  to  that  point  to  be  reflected  with 
a  change  of  sign  as  a  condensation.  The  reed  then  sends 
in  another  condensation,  and  the  two  condensations  are 
concordant.  These  running  into  the  pipe  meet  the  one 
reflected  from  the  distant  end  at  the  node  in  the  middle. 

The  disturbance  then  will  be  found  to  traverse  the  pipe 


204 


SOUND. 


twice  while  the  reed  executes  a  complete  double  vibration, 
or  the  length  of  the  pipe  is  half  the  wave-length  of  its 
fundamental  sound. 

This  constitutes  an  open  organ  pipe. 
It  has  a  node  near  its  middle  for  its 
gravest  tone.  If  a  stopped  pipe  and  an 
open  pipe  give  notes  of  the  same  pitch, 
the  open  pipe  is  twice  the  length  of  the 
closed  one. 

A  node  is  a  place  of  minimum  motion 
and  maximum  change  of  density ;  an 
antinode,  on  the  other  hand,  is  a  place 
of  maximum  motion  and  minimum 
change  of  density.  The  node  at  the 
middle  of  an  open  pipe  for  its  funda- 
mental tone  may  be  shown  by  means  of  a 
thin  stretched  membrane  on  Avhich  some 
fine  sand  is  strewn  (Fig.  90).  When  this 
is  lowered  into  the  pipe  by  means  of  a 
thread  it  will  buzz,  except  near  the  middle 
where  the  sand  ceases  to  be  agitated. 

153.  Relation  of  the  Overtones  to 
the  Fundamental  in  Open  Pipes  (V.,  II, 
121) .  —  Since  reflection  from  the  open  end 
of  a  pipe  changes  the  sign  of  the  con- 
densation but  not  of  the  motion,  a  wave 
twice  reflected  will  have  the  primitive 
sign,  and  will  accord  with  a  direct  wave  if  the  course 
traversed  by  it,  or  the  double  length  of  the  pipe  2Z,  con- 
tains a  whole  number  p  times  the  length  of  the  wave  X. 
The  condition  for  reenforcemerit  is  then 


Fig.  90. 


PHYSICAL    THEORY  OF  MUSIC. 


205 


or  since  X  =  VT=  — , 

n 

n=pl- 

This  formula  contains  all  the  laws  relating  to  open  organ 
pipes,  which,  together  with  those  relating  to  stopped  pipes, 
are  known  as  the  laws  of  Bernoulli.  They  were  estab- 
lished by  Daniell  Bernoulli  in  1762. 

When  p  =  1          the  formula  becomes 

X  =-22, 

or  the  length  of  the  pipe  is  half  the  wave-length  of  the 
fundamental  sound  in  air. 

When  p  is  made  successively  2,  3,  4,  etc., 

2  1 

X  =  Z,  X  =  -  Z,  X  =  ~  I,  etc., 

3  "2i 

or  the  wave-lengths  are  represented  by  the  series  1,  ^     -  , 

^j        O 

T  ,  p ,  etc.,  while  the  vibration-frequencies  are  proportional 
to  the  series  1,  2,  3,  4,  5,  etc.,  both  including  the  funda- 

A.  B  O  D  E  F 


F 

F 

V 

V 

F 



N 

» 



Jf 

V 



N 



N 

V 



N 

__»  -_ 

V 

V 



N 



N 

Y 

V 

Y 

N 

w 

Fig.  91, 


206  SOUND. 

mental.  The  overtones  are  due  to  a  division  of  the  pipe 
into  vibrating  segments  as  shown  on  the  left  of  Fig.  91. 
A  is  the  fundamental  with  an  antinode  V  at  each  end  and 
a  node  iVat  the  middle. 

The  first  overtone  adds  another  node  and  another  anti- 
node,  since  nodes  and  antinodes  must  alternate  and  suc- 
ceed each  other  at  equal  distances  apart.  The  half 
vibrating  segment  in  B  from  either  open  end  to  the 
adjacent  node  is  one-quarter  of  the  length  of  the  pipe  and 
half  as  long  as  for  the  fundamental  tone.  Its  frequency  is 
therefore  twice  as  great. 

For  the  second  overtone,  as  shown  in  (7,  still  another 
node  and  antinode  are  added,  the  half  segment  is  now 
reduced  to  one-third  its  primitive  length,  and  the  frequency 
is  three  times  that  of  the  fundamental.  The  next  over- 
tone would  require  four  nodes,  the  next  five,-  and  so  on. 

154.  Relation  of  Overtones  to  the  Fundamental  in 
Stopped  Pipes  (V.,  II,  123).  —  In  the  stopped  pipe  the 
reflection  at  the  closed  end  changes  the  sign  of  the  motion, 
while  the  reflection  at  the  mouthpiece  changes  the  sign  of 
the  condensation.  A  wave  twice  reflected  will  have  the 
sign  both  of  its  motion  and  its  condensation  reversed,  and 
it  will  accord  with  a  new  direct  wave  from  the  origin  if 

the  course  traversed,  2Z,  increased  by  -  ,   is    equal    to    p\. 

2i 

Whence 


or 

Therefore  n  =  (2p  -  1)  V  . 

When  p  =  1,  X  —  4£,  or  four  times  the  length  of  the  pipe. 


PHYSICAL    THEORY  OF  MUSIC.  207 

^ 

When  p  has  successive  values  2,  3,  4,  etc.,  X  =  -  Z,  X  = 

0 

-  /,  X  =  -  Z,  etc.,  in  succession,  or  the  wave-lengths  are  rep- 
o  7 

resented  by  the  series  1,  — ,  —    -  ,  etc., 

o    o    7 

and  the  vibration-frequencies  by  the  series 

1,  3,  5,  7,  etc., 
both  including  the  fundamental. 

The  overtones  of  stopped  pipes  are  due  to  a  division  into 
segments  as  represented  in  D,  E,  F,  Fig.  91. 

The  first  overtone  adds  one  node  and  one  antinode,  so 
that  the  half  segment  is  one-third  as  long  as  for  the  fun- 
damental, as  shown  in  E,  and  the  frequency  is  three  times 
as  great. 

For  the  second  overtone,  two  nodes  and  two  antinodes 
additional  to  those  of  the  fundamental  are  required.  The 
whole  pipe  is  therefore  divided  into  five  half  vibrating 
segments,  each  is  one-fifth  as  long  as  for  the  fundamental, 
and  the  frequency  is  five  times  as  great;  and  so  on. 

In  the  successive  internodal  spaces  the  motions  are 
always  in  opposite  directions,  or  of  opposite  sign.  After 
a  half  period  of  the  sound  produced,  these  motions  are 
all  again  equal,  but  have  changed  sign.  The  motions 
are  always  of  opposite  sign  on  the  two  sides  of  a  node. 
Also  at  any  instant  successive  nodes  are  affected  by 
alternative  condensations  and  rarefactions;  and  these  all 
change  their  signs,  or  exchange  places,  every  half-period 

T 

_  .     At  intermediate  instants  the   air  is  at   atmospheric 

pressure. 


208  SOUND. 

155.  Experimental  Verification. -- The  experimental 
verification  of  the  order  of  overtones  in  both  open  and 
stopped  pipes  is  readily  made  by  means  of  a  series  of 
diapasons  having  relative  vibration-rates  of  1,  2,  3,  4, 
5,  etc. 

One  must  also  be  provided  with  two  long  narrow  pipes, 
whose  fundamentals  are  in  unison  with  the  lowest  diapason, 
one  open  and  the  other  stopped.  On  such  long  pipes  it 
is  difficult  to  obtain  the  fundamental,  but  one  may  be 
assured  that  it  is  nearly  in  unison  with  the  diapason  by 
breathing  into  it  and  sounding  the  fork  at  the  same  time. 
Faint  beats  may  then  be  perceived  if  there  is  a  slight  dif- 
ference in  the  pitch.  If  the  pipe  is  the  open  one,  then  by 
blowing  slightly  harder  and  sounding  the  second  fork  an 
octave  higher  than  the  first,  audible  beats  will  again  be 
produced,  showing  that  the  first  overtone  has  twice  the 
frequency  of  the  fundamental.  The  second  overtone  will 
be  found  to  be  in  unison  or  to  beat  slowly  with  the  third 
fork,  the  next  overtone  with  the  fourth  fork,  and  so  on. 
The  partial  tones  of  the  open  pipe  are  thus  the  octave, 
the  octave  plus  the  fifth,  the  double  octave,  etc.,  of  the 
fundamental. 

Similar  experiments  made  with  the  stopped  pipe  will 
show  that  its  first  overtone  will  beat  slowly  with  the  third 
fork  of  the  series,  its  second  with  the  fifth,  etc.  The  par- 
tial tones  produced  by  it  are  therefore  an  octave  plus  a 
fifth,  two  octaves  plus  a  third,  etc.,  above  the  funda- 
mental. 

The  position  of  the  antinodes  may  be  found  by  the 
simple  device  of  piercing  the  side  of  the  pipe  with  small 
holes  at  the  points  where  the  antinodes  are  for  any  partic- 
ular overtone  selected.  The  pressure  at  the  antinode  is 
always  atmospheric.  An  opening  made  there  will  not 


PHYSICAL    THEORY   OF  MUSIC. 


209 


then  affect  the  sound,  while  the  pitch  will  change  if  an 
opening  be  made  at  any  other  point.  These  small  open- 
ings in  the  side  of  a  narrow  wood  pipe  can  be  covered  by 
turning  a  small  button.  Suppose  the  second  overtone  of 
an  open  pipe  is  blown.  The  division  into  segments  is 
shown  in  C  (Fig.  91),  and  there  is  an  antinode  at  one- 
third  the  length  of  the  pipe  from  either  end.  If,  then, 
either  button  be  turned  so  as  to  open  the  pipe  at  V,  no 
change  in  pitch  will  be  produced  for  this  overtone.  For- 
the  first  overtone  the  hole  may  be  at  the  middle  without 
affecting  the  sound. 

With  the  closed  pipe,  on  the  other  hand,  the  opening 
for  the  first  overtone  must  be  made  at  one-third  the  length 
of  the  pipe  from  the  closed  end,  and  for  the  second  over- 
tone one-fifth  or  three-fifths  from  the  closed  end,  E  and 
F  (Fig.  91). 

This  experiment 
demonstrates  that 
there  is  no  change  in 
pressure  at  the  anti- 
nodes.  Change  of 
pressure  occurs,  how- 
eve  at  all  o  t  h  e  r 
points,  and  especially 
at  the  nodes.  Koe- 
nig's  "  manometric 
flames  "  are  admira- 
ble for  illustrating 
this  phase  of  the 
phenomena  of  organ 
pipes.  At  the  proper 
points  in  the  side  of  a  pipe  holes  about  three  cms.  in 
diameter  are  covered  with  a  thin  diaphragm  of  gold- 


Fig,  93. 


Fig.  94. 


210  SOUND. 

beater's  skin,  or  rubber.  Over  this  is  fastened  a  small 
chamber  or  capsule,  into  which  illuminating  gas  is  ad- 
mitted. A  small  burner  is  attached,  and  the  flame 
is  examined  by  means  of  reflection  from  a  rotating 
mirror.  The  membrane  takes  the  motion  of  the  air  in  the 
pipe  and  communicates  it  to  the  gas  on  the  other  side 
of  it.  This  change  of  pressure  causes  the  gas  flame  to 
vibrate  in  unison  with  the  changes  of  pressure,  and  its 
image  in  the  rotating  mirror  is  a  serrated  band  (Fig.  92), 
which  represents  the  fundamental  tone.  If  the  pressure  is 
increased  so  as  to  produce  the  first  overtone  in  an  open 
pipe,  there  are  twice  as  many  tongues  of  flame  as  before 
(Fig.  93).  By  properly  regulating  the  pressure  both  the 
fundamental  and  the  first  overtone  may  be  produced  si- 
multaneously. The  appearance  of  the  flame  in  the  mirror 
is  then  as  represented  in  Fig.  94. 

156.  Kundt's  Experiment  (V.,  II,  159).  —  The  divis- 
ion of  a  resonant  pipe  into  segments  is  most  beautifully 
shown  by  means  of  a  glass  tube  about  two  cms.  in  diameter 
and  half  a  metre  long.  One  end  is  closed  and  a  common 
whistle  is  attached  to  the  other  (Fig.  95).  Within  the 


Fig.  95. 

tube  is  placed  a  little  lycopodium  powder,  or  better, 
amorphous  silica.  When  the  whistle  is  blown  the  powder 
gathers  itself  together  at  the  nodes  in  heaps,  and  at  the 
same  time  each  heap  is  divided  into  thin,  airy  segments 
by,  vertical  stratifications.  The  agitation  is  sufficient  to 
support  the  powder  in  opposition  to  gravity.  The  distri- 
bution of  the  powder  exhibits  the  stationary  waves  due  to 


PHYSICAL    THEORY  OF  MUSIO.  211 

the  superposed  direct  and  reflected  systems,  and  the  strati- 
fication shows  the  shifting  of  the  nodes  resulting  from  a 
lack  of  covibration  between  the  whistle  and  the  aerial  seg- 
ments. The  subdivision  of  the  pipe  changes  when  the 
pitch  of  the  whistle  changes  with  increase  of  pressure. 

Kundt  has  given  to  this  experiment  a  very  elegant  form, 
designed  to  compare  the  velocity  of  sound  in  air  and  other 
media.  His  apparatus  consists  of  a  long  glass  tube, 
closed  at  one  end  by  a  cork  furnished  with  a  stem,  which 
permits  of  slight  adjustment  by  forcing  in  or  withdrawing. 
Into  the  other  end  passes  a  rod  securely  clamped  at  its 
middle  and  terminated  in  the  interior  of  the  tube  by  a 
light  disk  of  a  diameter  slightly  less  than  the  tube.  The 
interior  of  the  tube  is  lightly  powdered  with  the  amorphous 
silica  or  fine-sifted  cork  filings.  When  the  rod  is  thrown 
into  longitudinal  vibration  by  friction  it  vibrates  precisely 
like  an  organ  pipe  open  at  both  ends,  and  giving  its 
fundamental  tone.  The  disk  on  the  inner  end  commu- 
nicates its  displacements  to  the  air  enclosed  in  the  tube, 
and  the  gaseous  column  tends  to  divide  into  segments  of 
such  length  that  they  will  all  vibrate  in  unison  with  the 
rod.  The  adjustment  for  unison  is  made  by  moving 
the  stopper  till  the  powder  gathers  into  small  detached 
heaps,  which  indicate  very  neatly  the  exactness  of  the 
adjustment.  The  column  of  air  is  then  an  exact  multiple 
of  half  wave-lengths,  and  the  distance  I  between  two 
adjacent  nodes  is  the  half  wave-length  of  the  sound  in 

air,  ~.     The  half  wave-length  in  the  solid  rod  is  its  length 

L.  These  distances  are  traversed  in  the  same  time,  and 
therefore 

I        v 


212  SOUND. 

the  ratio  of  the  velocity  of  sound  in  the  air  and  in  the 
solid.  Knowing  the  velocity  of  sound  in  air  at  the  tem- 
perature of  the  experiment,  the  ratio  gives  the  velocity  in 
the  solid,  and  this  is  connected  with  its  coefficient  of 

elasticity  by  the  formula  V  —  ,*  /—  ,  d  being  density  (117). 

Kundt  has  by  means  of  this  apparatus  confirmed  the 
theory  -of  von  Helmholtz  that  : 

1.  The  velocity  of  sound  in  a  tube  diminishes  with  the 
diameter,  or  the  length  of  the  tube  is  less  than  a  quarter 
of  the  wave-length  of  the  sound  considered. 

2.  The  diminution  is  greater  for  grave  sounds  than  for 
acute  ones. 

Kundt  has  also  verified  the  law  that  the  velocity  is  inde- 
pendent of  the  pressure  between  400  mm.  and  1760  mm. 
of  mercury. 

He  has  also  confirmed  the  law  of  the  variation  of 
velocity  with  temperature,  viz.,  that  velocity  is  propor- 


tional to  V  1  +  at. 

Similar  apparatus  serves  to  compare  the  velocity  of 
sound  in  different  gases. 

157.  Perturbations  at  the  Extremities  of  Pipes  (V., 
11,132).  —  Experiments  similar  to  those  described  con- 
firm the  indications  of  theory  that  the  internodal  distances 
are  constant  and  equal  to  the  half  wave-length  of  the 
sound  emitted,  except  at  the  open  ends,  and  particularly 
the  first  segment  next  to  the  mouth-piece.  Koenig  found 
for  the  eighth  sound  of  an  open  pipe,  that  is  for  the 
seventh  overtone,  the  following  lengths  in  millimetres  of 
half  segments  beginning  at  the  mouth-piece  : 

173,  315,  320,  314,  316,  312,  309,  271. 
The  mean  of  the  six  middle  ones  is  314  ;  the  first  is  less 


PHYSICAL    THEORY  OF  MUSIC.  213 

than  this  mean  by  141,  and  the  last  by  43.  The  length  of 
the  pipe  was  233  cms.  and  its  breadth  12  cms. 

The  length  of  a  pipe,  open  or  closed,  is  less  than  the  theo- 
retical length  of  -~  or  ^  for  its  fundamental  tone.  When 
the  pipe  gives  a  superior  tone,  the  internodal  distances  are 

always     -  with  the    exception  of   the   first  and  the  last. 

2 

These  variations  have  been  called  perturbations  at  the  ex- 
tremities. The  more  important  perturbation  is  the  one  at 
the  mouth-piece. 

At  the  free  extremity  of  the  open  pipe,  under  the 
influence  of  the  current  of  air  traversing  the  pipe,  the 
vibrating  column  is  prolonged  beyond  the  walls.  The  re- 
flection from  the  external  air  is  not  then  exactly  at 
the  plane  through  the  extremity  of  the  pipe,  but  a  little 
further  out,  and  the  condensation  at  this  plane  of  reflec- 
tion is  not  rigorously  zero.  Moreover,  there  are  probably 
multiple  reflections  at  the  ends  of  the  pipe  from  the  suc- 
cessive layers  of  air. 

It  is  not,  then,  the  length  of  the  pipe  itself,  but  this  length 
augmented  by  a  constant  quantity  £,  which  ought  to  be  an 
exact  multiple  of  the  half  wave-length  for  reenforcement 
by  an  open  organ  pipe.  So  also  for  a  closed  pipe  the 
sounds  energetically  reenforced  are  those  for  which  the 
length  of  the  pipe,  augmented  by  a  constant  quantity  Zx, 
is  an  uneven  multiple  of  a  quarter  of  a  wave-length. 
Wertheim  found  I1  =  0.746^,  where  R  is  the  radius  of  the 
pipe.  Other  investigators  have  found  other  values,  and 
the  whole  difficulty  remains  to  be  resolved. 

158.  Beats  due  to  Overtones  (A.  and  B.,  385).  — 
Beats  are  produced  not  only  between  two  notes  nearly  in 


214  SOUND. 

unison,  but  between  notes  whose  interval  is  approximately 
an  octave,  a  major  third,  a  fifth,  and  so  on.  These  von 
Helmholtz  attributes  to  the  overtones  associated  with  the 
fundamentals.  Thus  if  two  notes  have  vibration-frequen- 
cies n  and  2n  +  1,  then  the  first  overtone  of  the  lower 
will  be  due  to  2w  vibrations  per  second,  and  this  will  pro- 
duce one  beat  per  second  with  the  higher  note.  So  also 
if  two  notes  are  due  to  2n  +  1  and  3n  vibrations  per  second 
respectively,  then  the  second  overtone  of  the  first  will  be 
due  to  6n  +  3,  and  the  first  overtone  of  the  second  to  6n 
vibrations  per  second,  giving  three  beats  per  second, 
though  the  interval  is  otherwise  indistinguishable  from 
a  fifth.  Combinations  of  such  vibrations,  obtained  me- 
chanically by  Koenig,  show  periodic  variations  of  ampli- 
tude corresponding  with  the  beats. 

Again,  the  interval  between  the  fundamentals  may  be 
exact,  but  the  overtones  may  be  partial  tones,  and  so  not 
exact  multiples  of  the  fundamentals.  Such  is  the  case 
with  tuning-forks,  and  beats  are  sometimes  heard  between 
their  overtones  of  the  same  order. 

159.  The  Quality  of  Sound  (H.,  106  (113)  ;  Z.,  341; 
V.,  II,  292). --Two  of  the  essential  characteristics  of 
musical  sounds  have  already  been  considered,  viz.,  pitch 
and  loudness  or  intensity.  But  there  is  a  third  important 
difference  between  musical  sounds,  known  as  their  quality 
or  timbre.  We  easily  recognize  that  one  sound  differs 
from  another  not  only  in  being  more  acute  or  grave, 
louder  or  softer,  but  also  in  respect  to  the  character  of  the 
sound  itself.  We  have  no  difficulty  in  distinguishing 
the  notes  of  the  violin  from  those  of  the  piano,  even 
though  they  are  of  the  same  fundamental  pitch  and  loud- 
ness.  In  the  same  way  we  learn  to  distinguish  one  voice 


PHYSICAL    THEORY  OF  MUSIC.  215 

from  another  in  speech  as  well  as  in  song,  even  when 
somewhat  distorted  in  transmission  by  the  telephone,  or 
when  reproduced  by  the  phonograph.  The  same  musical 
instrument  may  emit  tones  with  marked  differences  de- 
pending upon  the  player ;  and  even  the  untrained  musical 
ear  can  readily  distinguish  between  the  character  of  the 
music  produced  by  different  instruments  of  the  same 
class.  The  tones  of  a  modern  violin  are  far  inferior  to 
those  emitted  by  an  old  Stradivarius,  for  example  ;  and 
different  players  evoke  different  tones  from  a  Stradivarius. 
All  these  differences,  not  assignable  to  pitch  or  loudness, 
are  included  under  the  term  quality. 

If  we  seek  for  the  physical  basis  of  the  three  character- 
istics of  musical  sounds,  we  know  that  pitch  depends 
upon  the  wave-length,  and  loudness  upon  the  amplitude 
of  vibration ;  quality  must  therefore  depend  upon  the  only 
other  physical  difference  between  aerial  sound-waves,  viz., 
their  vibrational  form.  By  form  is  meant  the  law  accord- 
ing to  which  the  velocities  or  displacements  of  the  air  parti- 
cles change  from  point  to  point  along  the  path  of  the  wave. 
This  may  be  expressed  either  graphically  or  by  means  of 
a  mathematical'  equation.  Let  the  two  upper  curves  in 
Fig.  96  represent  two  simple  harmonic  motions  in  the  same 
medium  with  periods  as  two  to  three ;  the  amplitudes  are 
the  same.  Their  resultant  may  be  found  by  adding 
together  corresponding  ordinates  with  their  proper  sign. 
It  is  represented  by  the  heavy  line  below.  The  complete 
vibrational  form  which  is  repeated  over  and  over  is  not 
quite  all  shown  in  the  figure.  It  recurs  with  three  wave- 
periods  of  the  first  and  two  of  the  second  component 
motions. 

It  will  be  remembered  that  according  to  Fourier's 
theorem,  any  periodic  vibration  admits  of  resolution  into  a 


216 


SOUND. 


fundamental  or  prime  simple  harmonic  motion,  and  har- 
monics having  frequencies  represented  by  exact  multiples 
of  the  prime.  Any  change  in  the  vibrational  form  of  such 
a  wave,  not  affecting  pitch  or  loudness,  must  then  be  due 


x 


TB'-CB 


/ 


\ 


\ 


A  B  C 


\ 


Fig.  96. 

to  some  change  in  the  components  higher  than  the  funda- 
mental, or  in  the  overtones. 

If  we  consider  partial  tones  arising  from  the  several 
modes  of  vibration  which  a  sonorous  body  may  execute 
simultaneously,  some  of  these  may  harmonize  with  the 
prime  or  be  harmonic  partials,  while  others  are  inharmonic. 
Those  of  a  stretched  string  are  in  general  harmonic,  while 


PHYSICAL    THEORY  OF  MUSIC,  217 

those  of  bells,  plates,  and  tuning-forks  are  inharmonic.  But 
the  motions  of  the  air  particles,  conveying  all  these  sounds 
to  the  ear,  must  be  such  as  to  represent  all  the  component 
sounds,  since  at  any  point  there  can  be  at  one  instant  only 
one  definite  density  and  a  definite  velocity  in  one  direction. 
Each  air  particle  has  therefore  impressed  upon  it  the 
motions  representing  the  several  partial  sounds,  and  its 
motion  is  the  resultant  of  all  of  these.  The  ear  possesses 
the  marvellous  property  of  analyzing  this  complex  motion 
into  its  constituents  and  of  thus  selecting  out  the  component 
tones  which  enter  into  the  complex  melange.  Now  the 
vibrational  form  of  the  complex  resultant  wave  depends 
upon  the  presence  of  the  overtones,  which  impress  modifi- 
cations upon  the  fundamental ;  and  the  musical  quality  of 
the  tone  is  in  the  same  way  determined  by  the  presence  of 
the  overtones  associated  with  the  fundamental.  The  con- 
clusion of  von  Helmholtz,  derived  both  from  the  analysis 
and  the  synthesis  of  musical  sounds,  is  as  follows  :  "  The 
quality  of  the  musical  portion  of  a  compound  tone  depends 
solely  on  the  number  and  strength  of  the  partial  tones,  not 
on  their  differences  of  phase." 

The  vibrational  form  of  a  sound-wave  depends  upon  the 
number,  the  order,  the  relative  intensity,  and  the  relative 
phase  of  the  overtones  associated  with  the  fundamental. 
But  the  quality  of  a  musical  sound  being  defined  by  the 
number  and  the  intensity  of  the  simple  sounds  into  which 
it  is  decomposable  in  accordance  with  Fourier's  theorem, 
the  phase  does  not  appear  to  be  able  to  intervene.  Koenig 
has  tried  to  show  that  phase  difference  among  the  overtones 
does  produce  a  change  in  quality.  His  instruments  for  this 
purpose  are  called  wave  sirens.  Physicists  are  divided  in 
opinion  respecting  the  validity  of  the  conclusion  which 
Koenig  draws  from  his  experiments.  A  change  in  phase 


218  SOUND. 

certainly  appears  to  produce  a  marked  change  in  the  quality 
of  the  resulting  sound ;  but  the  objection  is  raised  that 
Koenig's  method  of  producing  the  component  tones  does 
not  insure  such  simplicity  of  the  sounds  combined  as  one 
requires  in  order  to  admit  the  interpretation  which  he 
gives  to  these  interesting  phenomena. 

160.  Resultant  Tones  (V.,  II,  253  ;  Z.,  322;  H.,  229 
(253) ;  D.,  438).  — When  two  sufficiently  intense  sounds 
of  frequencies  n  and  n',  differing  less  than  an  octave,  are 
produced  together,  they  give  rise  to  a  third  sound,  the 
vibration-rate  of  which  is 

N=n'  —  n. 

These  are  called  resultant  tones.  They  were  discovered 
by  Sorge  in  1740,  and  independently  by  Tartini  in  1754 ; 
they  are  therefore  often  called  Tartini's  tones.  Tartini 
gave  to  them  the  name  of  third  tones. 

If  we  sound  together  two  forks  c"  and  g",  whose  inter- 
val is  a  fifth  and  whose  frequencies  are  512  and  768,  we 
obtain  a  distinct  tone  c1  with  a  frequency  of  768  —512  = 
256.  So  also  c"  and  e",  with  frequencies  512  and  640, 
give  as  a  resultant  tone  640  —  512  =  128,  or  c. 

These  tones  have  been  called  differential  tones  by  von 
Helmholtz,  because  their  frequency  is  the  difference  of  the 
frequencies  of  the  two  tones  from  which  they  arise.  Other 
resultant  tones,  called  summational  tones,  were  discovered 
by  von  Helmholtz.  Their  frequency  is  the  sum  of  the 
frequencies  of  the  tones  producing  them,  or 
N=  n'  +  n. 

Thus  c?  and  g',  frequencies  256  and  384,  will  produce 
together  e"  of  640  vibrations  per  second,  and 
256  +  384  =;  640. 


PHYSICAL    THEORY  OF  MUSIC.  219 

Summational  tones  are  much  feebler  than  differential 
tones  — •  a  fortunate  circumstance,  since  they  are  mostly 
inharmonic. 

Resultant  or  combinational  tones  may  be  attributed  to 
two  causes.  The  first  one  is  in  the  external  air,  and  the 
other  has  its  origin  in  the  drum  of  the  ear. 

When  two  disturbances  from  separate  sources  are  super- 
posed, we  have  assumed  that  the  amplitude  of  the  resultant 
disturbance  at  any  point  is  the  sum  of  the  two  separate 
disturbances.  This  is  true  only  so  long  as  the  displace- 
ments are  very  small.  The  amplitude  of  the  compound 
oscillation  for  greater  disturbances  falls  short  of  the  sum 
of  the  amplitudes  of  the  components.  This  is  equivalent 
to  the  introduction  of  another  vibration  whose  frequency  is 
that  of  the  differential  tone.  Particular  instruments  give 
powerful  combinational  tones  when  the  same  mass  of  air 
is  violently  agitated  by  two  simple  tones  simultaneously. 
This  is  the  case  with  a  siren  in  which  two  or  more  rows 
of  holes  are  blown  from  the  same  wind-chest ;  and  with 
the  harmonium,  in  which  openings,  closed  and  opened 
rhythmically  by  the  tongues  or  reeds,  communicate  with  a 
common  wind-chest.  This  objective  portion  of  combina- 
tional tones  can  be  reenforced  by  resonators. 

But  the  vibrations  corresponding  to  combinational  tones 
may  exist  in  the  parts  of  the  ear  without  any  objective 
existence  in  the  external  air.  The  construction  of  the  ex- 
ternal ear  is  peculiarly  favorable  to  the  generation  of  com- 
binational tones.  Its  external  convex  radial  fibres  undergo 
a  greater  change  of  tension  when  they  oscillate  with  a 
moderate  amplitude  inwards  than  Avhen  the  oscillation  is 
outwards.  "  Under  these  circumstances  deviations  from 
the  simple  superposition  of  vibrations  arise  for  very  much 
smaller  amplitudes  than  is  the  case  where  the  vibrating 


220  SOUND. 

body  is  symmetrically  constructed  on  both  sides."    Sounds 
so  produced  cannot  be  reeiiforced  by  resonators. 

161.  Dissonance  due  to  Beats  (V.,  II,  3O6  ;  Z.,  422  ; 
H.,  251(277)).  —  The  theory  of  von  Helmholtz  to  explain 
dissonance  proceeds  from  the  fact  that  beats  produce  in  the 
nerves  of  audition  an  intermittent  excitation,  and  this  is 
disagreeable  as  are  all  excitations  of  this  kind,  like  that 
produced  by  a  fluttering  light.  According  to  von  Helm- 
holtz beats  become  most  disagreeable  at  33  per  second.  At 
132  per  second  they  cease  to  be  perceptible.  Further,  tin; 
distinctness  of  the  beats  and  the  harshness  of  the  interval 
do  not  depend  solely  on  the  absolute  difference  in  the 
number  of  vibrations ;  they  are  dependent  also  upon  the 
ratio  of  these  numbers,  or,  as  we  commonly  say,  the  magni- 
tude of  the  interval.  If  we  could  disregard  their  magnitude, 
all  the  following  intervals,  which  give  33  beats  each,  would 
be  equally  rough : 

The  semitone b',  495  —  c",  528. 

The  whole  tone c7,  264  -  d',  297. 

The  minor  third      ....  e,  165  -  g,  198. 

The  major  third      ....  c,  132  —    e,  165. 

The  fourth #,    99  -  -    <?,  132. 

The  fifth .  <7,    66  -a,    99. 

But  we  find  the  deeper  intervals  more  and  more  free 
from  roughness.  If  the  two  sounds  are  sufficiently  sepa- 
rated in  the  musical  scale,  the  audition  fibres  simultaneously 
affected  by  the  two  sounds  vibrate  too  feebly  for  the  beats 
to  be  appreciable.  For  a  given  frequency  of  beats,  there- 
fore, the  harshness  of  the  effect  increases  with  the  -nearness 
of  the  notes  on  the  musical  scale. 

Thus  far  we  have  supposed  the  sounds  producing  disso- 


PHYSICAL    THEORY  OF  MUSIC. 


221 


nance  are  two  simple  tones.  But  musical  sounds  are  always 
accompanied  by  harmonic  partials,  the  importance  of  which 
is  apparent  upon  inspection  of  the  several  intervals  with 
their  partials  below : 

9 


Twelfth 



Fifth      . 
Fourth 

'.     .     .     2     ' 

3 
.   3     6 

1234567 


9     1C 


Major  sixth    . 


123456789     16 
123456789     10 


2          4          6  8          1G 

123456789     10 

I  I  I 

369 

4     6     8     10     12     14     16     18     20 

I  I  I 

6        9          12        15         18 

9    12     15     18     21     24     27     30 


4     8       12         16     20         24         28 
3     6     9     12     15     18     21     24     27     30 


5         10         15         20  25         30 

Major  third    ..  4     8     12     16     20     24     28     32     36     40 

I  I 

5     10     15       20         25     30     35        40 

The  larger  the  number  of  pairs  of  upper  partials  in  any 
interval  near  enough  to  each  other  on  the  scale  to  produce 
distinct  beats,  the  greater  will  be  the  dissonance  of  the  in- 
terval. If  such  beating  pairs  are  altogether  wanting,  or 
have  so  little  intensity  that  their  effect  is  negligible,  the 
interval  is  a  consonant  one. 

Four  distinct  groups  can  be  readily  distinguished. 

1.  Unison,  the  octave,  and  the  twelfth,  to  which  also 
may  l>e  added  the  double  octave.  The  consonance  here  is 


222  SOUND. 

perfect.     The  prime  and  partials  of  the  upper  tone  all  coin- 
cide with  partials  of  the  lower. 

2.  The  fifth  and  the  fourth.     These   may  also  be   re- 
garded as  perfectly  consonant,  because  they  may  be  used 
in  all  parts  of  the  scale  without  any  disturbance  of  the 
harmony.     The  fourth  is  the  less  perfect  consonance.     It 
owes  its  superiority  in  musical  practice  simply  to  its  being 
the  defect  of  a  fifth  from  an  octave. 

3.  The  major  sixth  and  major  third.     These  are  called 
medial  consonances.     They  were  considered  imperfect  con- 
sonances by  old  writers  on  harmony.     The   coincidences 
among  the  upper  partials    are    few   in  number,  and   the 
beating  pairs  are  more  numerous  than   in  the    preceding 
intervals. 

4.  The  minor  third  and  the  minor  sixth.     These  are  im- 
perfect consonances.    Coincident  pairs  of  partials  are  almost 
or  quite  wanting,  and  the  pairs  near  enough  to  beat  are 
either  the  semitone  16  :  15  or  the  semitone  25  :  24.    These 
partials,  on  which  the  definition  of  the  interval  depends, 
are  often  absent,  or  the  imperfection  in  the  interval  pro- 
duced by  them  is  slight. 

162.  The  Musical  Importance  of  Resultant  Tones 
(V.,  II,  256).  —  Resultant  tones  play  an  important  rdle  in 
music  by  reason  of  the  influence  which  they  exert  on  con- 
sonance. 

Consider,  for  example,  a  perfect  major  chord  in  which  the 
vibration-frequencies  have  the  relation 
4,  5,    6,    8, 

1    6     3     2 

or  '  4'   2'   ' 

The  resultant  difference  tones  are 
1     1     3     1 
4'   2'   4'     ' 


PHYSICAL    THEORY  OF  MUSIC.  223 

or  the  double  lower  octave  of  the  fundamental,  the  lower 
octave  of  the  same,  the  octave  below  the  fifth,  and  the 
fundamental  sound ;  all  of  these  reenforce  the  primary 
sounds  and  particularly  the  fundamental.  The  differential 
tones  all  strengthen  the  consonance. 

To  illustrate  numerically,  let  the  major  chord  consist  of 
c./,  e',  g',  c".  These  with  their  differential  tones  may  then 
be  written  as  follows  : 

C  c  g  c'  e'  g'  c" 

64    128    192    256    320    384    512 

1x64      2x64     3x64     4x64     5x64     6x64     8x64 

The  entire  series  consists  of  C  with  its  harmonic  partials 
up  to  the  eighth  in  order,  with  the  exception  of  7  x  64, 
which  would  mar  the  consonance.  The  differential  tones 
contain  no  discordant  elements. 

In  the  perfect  minor  chord  the  frequencies  are 
10,  12,  15,  20, 

1     6     3     9 
1,    ^   3,   2. 

The  differences  are 

1      3^     1     4     -, 
5'  10'  2'  5' 

These  are  the  double  octave  below  the  grave  major 
third ;  the  double  lower  octave  of  the  minor  third ;  the 
octave  below  the  fundamental ;  and  finally  the  fundamen- 
tal itself.  A  new  sound  appears  here,  the  grave  major 
third,  which  is  dissonant  with  the  fifth.  This  explains  the 
indecisive,  pathetic  character  of  the  minor  chord. 

163.  Musical  Pitch  (D.,  385  ;  Z.,  75).  —  A  progressive 
change  in  the  absolute  pitch  of  the  notes  of  the  arbitrary 
scale  employed  by  musicians  has  been  going-  on  for  many 


224  SOUND. 

years.  Attempts  have  been  made  in  recent  times  to  arrest 
this  upward  tendency,  and  Avith  some  success. 

In  the  time  of  Handel  the  middle  A  (V)  of  the  piano,  or 
the  second  string  of  the  violin,  was  made  by  424  vibrations 
a  second ;  while  the  organ  pitch  in  England  in  the  middle 
of  the  eighteenth  century  was  as  low  as  a1  =  388  vibra- 
tions. At  Paris  in  1700  middle  A  was  405  ;  later  on,  425 ; 
in  1855,  440 ;  and  in  1857,  448  vibrations  per  second.1 

Quite  recently  the  pitch  of  the  Theatre  de  la  Scala  in 
Milan  was  451.5,  and  that  of  the  Co  vent  Garden  Theatre, 
London,  455.  Modern  concert  pitch  has  risen  as  high  as 
460  for  a'.  The  sharping  of  the  notes  gives  a  certain  bril- 
liancy to  the  music  of  orchestral  instruments,  but  makes  a 
great  demand  upon  the  powers  of  concert  singers,  particu- 
larly as  much  modern  music  is  written  upon  high  notes. 

"  It  was  necessary  to  find  a  remedy  for  so  grave  an  in- 
convenience, and  therefore  an  international  commission 
fixed  as  the  normal  pitch  a  tuning-fork  giving  435  vibra- 
tions per  second."  This  corresponds  with  the  recommen- 
dation of  the  Paris  Academy  of  Sciences. 

The  German  Society  at  Stuttgart  in  1834  recommended 
a'  =  440  as  the  standard  pitch.  This  makes  c"  —  528  vibra- 
tions per  second.  Since  528  is  22  times  24,  by  multiplying 
the  series  representing  the  vibration-rates  of  the  perfect 
diatonic  scale,  24,  27,  30,  32,  36,  40,  45,  48,  by  11,  we 
obtain  the  vibration-frequencies  of  the  notes  of  the  natural 
gamut  of  C  corresponding  to  this  standard  of  pitch. 

For  scientific  purposes  c'  is  taken  as  256  vibrations  per 
second.  Since  this  is  the  eighth  power  of  2,  any  power 
of  2  will  express  the  vibration-frequency  of  C  in  some 
octave,  according  to  this  standard.  On  the  same  standard 
a?  is  426|. 

1  Blaserna's  Theory  of  tiound,  p.  70. 


PHYSICAL     THEORY    (> /•'    MUStC. 

164.  Limits/of  Pitch  employed  in  Music  (BL,  67;  B., 

243). In  ther  modern  pianoforte  of  seven  octaves  the  bass 

A  corresponds  to  about  27.5  vibrations  per  second,  the 
highest  A  to  3480.  Taking  into  account  the  slight  vari- 
ations from  standards  in  tuning,  the  notes  employed  on  the 
piano  of  seven  octaves  range  between  27  and  3500  vibra- 
tions per  second.  Some  pianos  go  as  high  as  the  seventh 
C  with  4200  vibrations  per  second,  but  such  high  notes  arc 
shrill  and  lacking  in  sweetness.  The  practical  limits  are 
about  27  and  4000. 

On  the  organ  the  deepest  note  is  the  C  of  16  vibrations 
per  second  given  by  the  33-foot  open  pipe.  Its  wave-length1 

in  air  at  normal  temperatures  is  about  ———21.5  metres 

or  70.5  feet.  The  highest  note  is  the  same  as  the  highest 
A  of  the  piano,  which  is  the  third  octave  above  a'  =  435.  It 
is  made  therefore  by  435  x  23  =  3480  vibrations  per  second. 

The  well-developed  voice  of  a  singer  embraces  about  two 
octaves.  The  voice  of  woman  is  represented  by  about 
twice  as  many  vibrations  per  second  as  that  of  man.  The 
lowest  note  of  the  voice,  not  including  certain  exceptional 
cases,  is  C  of  65  vibrations.  The  entire  range  is  included 
between  this  note  and  c'"  of  1044  vibrations  per  second, 
the  two  higher  octaves  belonging  to  woman.  The  highest 
exceptional  voice  appears  to  be  that  of  Bastarclella.  whom 
Mozart  heard  at  Parma  in  1770.  Bastardella's  voice  had  a 
range  of  three  and  a  half  octaves  and  went  up  nearly  to 
2000  vibrations. 

The  limits  of  hearing  far  exceed  those  of  the  voice  or  of 
music.  They  are  about  16  for  the  lowest  note,  and  from 
38,000  to  40,000  for  the  highest.  The  limits  are  some- 
what different  for  different  persons ;  but  we  may  say  that 
sonorous  vibrations  lie  between  16  and  40,000  per  second. 


226  LIGHT. 


LIGHT. 


CHAPTER    VIII. 

NATURE    AND    PROPAGATION    OF    LIGHT. 

165.  Nature  of  Light.  —  The  sensation  of  light  is  due 
to  a  mechanical  action  upon  the  extension  of  the  optic 
nerve,  forming  the  sensitive  surface  of  the  retina.  The 
undulatory  theory  of  light,  now  universally  accepted,  as- 
signs this  action  to  a  disturbance  propagated  from  its 
source  by  a  wave-motion  in  the  universal  medium  called 
the  ether.  This  process  of  radiation,  as  it  is  called,  is 
clearly  a  process  of  transference  of  energy  through  the 
ether;  and  this  transfer  is  accomplished  by  periodic  dis- 
turbances in  the  medium  which  follow  the  laws  of  wave- 
motion.  These  disturbances,  according  to  the  theory  of 
Maxwell,  which  has  been  confirmed  by  the  remarkable 
experiments  of  Hertz,  are  electro-magnetic  phenomena. 
Light  objectively  "  now  takes  its  place  alongside  of  elec- 
tric phenomena,  as  but  one  of  the  forms  of  energy  asso- 
ciated with  that  wonderful  kind  of  matter  provisionally 
called  the  ether  "  (Tait). 

The  existence  of  the  ether  is  a  necessary  consequence  of 
licenser's  discovery  in  1676  of  the  finite  speed  of  light. 
For  the  transmission  of  light  is  the  transmission  of  energy; 
and  a  medium  of  transmission  is  a  necessary  postulate  as 
the  repository  of  this  energy  during  the  time  of  transmission. 


NATURE  AND   PROPAGATION  OF  Li&HT.  227 

The  earth  receives  energy  from  the  sun ;  and  as  something 
over  eight  minutes  are  consumed  in  its  transit  across  the 
intervening  space,  we  are  forced  to  seek  for  the  vehicle  by 
which  it  is  conveyed.  Only  two  methods  of  the  propa- 
gation of  energy  are  known  to  us,  and  no  other  method 
seems  now  possible  or  conceivable.  Energy,  according  to 
our  present  knowledge,  is  always  associated  with  matter, 
so  that  matter  has  been  defined  as  the  vehicle  of  energy. 

One  of  the  methods  by  which  matter  conveys  energy 
to  a  body  at  a  distance  is  that  of  a  projectile.  The  bullet 
carries  its  energy  of  motion  from  the  gun  to  the  mark 
across  the  intervening  space.  This  is  Newton's  corpusoo> 
lar  theory  of  light.  He  imagined  the  light-giving  body 
projecting  minute  particles,  or  corpuscles,  through  space ; 
and  that  these,  entering  the  eye,  excitecT vision  by  impact 
upon  the  retina. 

The  other  method  of  propagation  requires  a  continuous 
medium,  and  the  energy  is  handed  along  from  particle  to 
particle  as  an  undulation.  In  this  way  energy  is  conveyed 
in  sound  and  by  water-waves  across  the  surface  of  the  sea. 
In  the  first  method  the  energy  remains  with  the  matter 
transmitting  it  from  start  to  finish ;  in  the  second  it  is 
passed  along  from  particle  to  particle  by  the  series  of 
operations  which  transmit  the  wave-motion.  According 
to  this  latter  theory  a  luminous  body  is  the  centre  or 
source  of  a  disturbance  in  the  ether,  which  propagates  it 
in  waves  through  space.  They  travel  with  the  velocity  . 
of  light,  and  entering  the  eye  excite  the  sense  of  vision. 

Now  both  of  these  methods  involve  the  element  of  time, 
and  conversely  the  existence  of  the  time  element  appears 
to  limit  the  transmission  to  these  two  methods.  But  New- 
ton's corpuscular  theory  failed  because  of  its  complexity 
and  the  crucial  test  applied  to  it  by  Foucault  (Art.  172)  ; 


LIGHT. 

there  remains  then  only  the  undulatory  theory  and  the 
ethereal  medium. 

Physical  optics  consists  in  a  study  of  wave-motion  propa- 
gated through  the  ether  in  accordance  with  the  principle  of 
Huygheiis.  It  is  an  inquiry  into  the  physical  processes 
concerned  in  the  transmission  of  light  and  in  the  phenomena 
of  reflection,  refraction,  interference,  and  dispersion.  It 
seeks  to  account  for  them  entirely  by  the  application  of 
dynamical  principles.  Greometrical  optics,  on  the  other 
hand,  is  confined  to  the  exposition  of  those  facts  which  can 
be  studied  by  the  aid  of  the  simple  geometrical  considera- 
tions involved  in  the  laws  of  reflection,  refraction,  and  the 
transmission  through  isotropic  media  in  right  lines  or  rays, 
without  any  inquiry  into  the  nature  of  light  itself. 

In  this  elementary  treatment  of  the  subject  it  will  best 
suit  our  purpose  to  adhere  exclusively  neither  to  the  one 
method  nor  to  the  other,  but  to  make  use  of  either  the 
physical  or  the  geometrical  method  according  as  it  may 
best  serve  for  the  simple  exposition  of  fundamental 
principles. 

166.  The  Rectilinear  Propagation  of  Light  (V.,  II, 
311  ;L.,14;  T.,  24).  —  It  is  a  fact  of  common  experience 
that  in  a  uniform  medium,  possessing  the  same  properties 
in  all  directions,  light  travels  in  straight  lines  or  rays.  If  a 
source  of  light,  as  a  candle,  is  concealed  from  the  eye  by  an 
opaque  screen,  it  suffices  to  move  the  screen  so  as  to  un- 
•  the  right  line  connecting  the  candle  and  the  eye  in 
order  that  the  light  may  be  seen.  So  no  objects  can  be  seen 
through  a  straight  tube  except  those  situated  in  the  direc- 
tion of  the  axis  produced. 

But  rectilinear  propagation  is  confined  to  media  techni- 
cally described  as  both  homogeneous  an.l  isotropic.  In  other 


NATURE  AND   PROPAGATION  OF  LIGHT.  229 

words  the  wave-surfaces  must  be  truly  spherical  about  the 
source  as  a  centre.  In  such  a  medium  the  disturbance  at 
any  point  is  due  almost  entirely  to  the  motion  transmitted 
along  a  perpendicular  dropped  upon  the  wave-surface  in 
some  prior  position.  The  secondary  waves  transmitted 
from  all  other  points  of  the  surface,  as  new  centres  of  dis- 
turbance, neutralize  one  another  along  this  perpendicular 
on  account  of  the  excessive  minuteness  of  the  wave-lengths 
of  all  radiations  exciting  vision.1 

But  in  media  of  unequal  speed  of  transmission  in  differ- 
ent directions  the  wave-surfaces  are  not  spherical,  a  ray  of 
light  is  no  longer  necessarily  at  right  angles  to  the  wave- 
surface,  and  light  may  be  transmitted  in  curved  lines  if  the 
properties  of  the  media  vary  from  point  to  point. 

If  we  look  along  a  hot  bar  or  over  a  heated  surface 
objects  beyond  appear  distorted.  So  when  the  sun  shines 
on  a  hot  stove,  the  rising  currents  of  heated  air  produce 
the  appearance  of  images  or  shadows  on  a  white  wall 
beyond.  The  air  is  irregularly  expanded,  the  medium  is 
not  homogeneous,  and  the  light  no  longer  moves  in  straight 
lines.  To  this  lack  of  homogeneity  are  due  the  phenomena 
of  mirage,  the  duplication  of  images  of  a  distant  object 
seen  through  an  atmosphere  unequally  heated  above  and 
below,  and  the  twinkling  of  the  stars.  So  the  rectilinear 
propagation  of  light  is  limited  to  media  having  uniform 
optical  properties  in  all  directions  and  without  variations 
from  point  to  point. 

167.  Images  produced  by  Small  Apertures  (T.,  30; 
V.,  II,  315).  —  If  light  from  a  luminous  surface  pass 
through  a  small  opening  of  any  shape  in  an  opaque  screen 
and  fall  upon  a  white  surface  at  some  distance,  it  will  form 

1  Preston's  Theory  of  Light,  p.  54. 


230 


LIGHT. 


n 


Fig.  97. 


an  inverted  image  of  the  source  (Fig.  97).  Not  only  is 
this  image  EF  inverted,  but  it  is  perverted.  By  making  a 

small  opening 

A  B  in  the  shutter 

of  a  darkened 
room  an  image 
of  outside  ob- 
jects, brightly 
illuminated  by 
the  sun,  may  be 
obtained  upon 
a  white  screen 

held  at  a  suitable  distance.  Not  only  is  there  inversion, 
slm\vii  by  objects  being  depicted  lowe«r  the  higher  they  are, 
but  if  the  image,  vieAved  from  the  side  of  the  screen  toward 
the  opening,  be  imagined  turned  round  in  its  own  plane  so 
as  to  make  it  erect,  it  will  be  found  that  the  right  side 
of  the  object,  as  we  look  toward  it,  corresponds  with 
the  left  of  the  image.  If  the  screen  is  translucent  and  the 
image  is  viewed  from  behind,  it  will  then  be  inverted, 
but  not  perverted.  The  right  hand  may  be  considered  the 
perverted  image  of  the  left.  An  image  in  a  plane  mirror 
is  perverted,  but  not  inverted. 

Each  point  of  the  object  (Fig.  97)  is  the  vertex  of  a  cone 
of  rays  passing  through  the  aperture  and  forming  an  image 
of  it  on  the  screen.  These  images  will  be  symmetrically 
placed  with  reference  to  the  points  emitting  the  light,  and 
consequently  by  their  superposition  will  form  a  figmv  of 
the  same  outlines  as  the  luminous  object.  Since  a  smaller 
number  of  these  images  of  the  opening  overlap  near  the 
edges  of  the  image  on  the  screen,  the  edges  of  the  picture 
will  be  Ic'ss  bright  than  the  other  portions.  Moreover, 
the  sharpness  of  the  image  diminishes  as  the  opening  is 


NATURE  AND   PROPAGATION  OF  LIGHT.  231 

made  larger ;  all  the  images  of  the  opening  become  larger, 
and  with  a  large  opening  their  superposition  no  longer 
produces  a  picture  resembling  the  object  but  rather  the 
opening.  The  insertion  of  a  converging  lens  in  the  open- 
ing not  only  permits  of  a  larger  aperture,  thus  increasing 
the  illumination,  but  improves  the  definition  of  the  image 
by  causing  the  light  emanating  from  any  point  of  the 
object  to  come  to  a  focus  at  the  corresponding  point 
of  the  image  with  minimum  overlapping  of  neighboring 
images. 

A  particular  case  of  the  preceding  phenomenon  is  fur- 
nished by  the  light  of  the  sun,  sifting  through  the  foliage 
of  trees  and  tracing  circular  or  elliptical  images  on  the 
ground.  During  solar  eclipses  these  spots  change  into 
crescents,  which  are  the  more  pronounced  as  the  eclipse 
is  more  complete. 

168.  Theory  of  Shadows  (T.,  26 ;  L.,  15  ;  V.,  II,  311).— 
An  optical  shadow  is  the  region  from  which  light  is  wholly 
or  in  part  cut  off  by  an  opaque  body.  If  the  body  is  in  the 
vicinity  of  a  luminous  point  its  illuminated  area  is  limited 
by  the  curve  of  contact  of  a  cone  starting  from  the  point 
and  circumscribed  about  the  opaque  body  AB  (Fig.  98). 
The  portion  of  the  space  within  this  cone  of  shadow  and 
beyond  the  curve  of 
contact  is  entirely  in 
obscurity. 

But  when  the  lumi- 
nous source  has  sensi- 
ble dimensions,  then 
outside  of  the  total 
shadow  or  umbra,  and 
surrounding  it,  is  a  region  called  the  partial  shadow  or 


232  LIGHT. 

penumbra.  This  is  limited  by  the  double  cone  with  its  apex 
between  the  luminous  and  the  opaque  bodies  (Fig.  99). 

This  region  of  par- 
tial shadow  forms  the 
transition  from  com- 
plete obscurity  to 
the  full  light.  The 
smaller  the  angular  di- 
mensions of  the  source 

Fig.  99.  a  n  d    t  h  e    nearer    the 

screen  CD  to  the  opaque 

body,  the  narrower  will  be  the  penumbra.  Only  a  portion 
of  the  luminous  body  is  visible  to  an  eye  situated  within 
the  penumbra. 

Solar  eclipses  are  produced  by  some  portioi}  of  the 
earth's  surface  passing  through  the  shadow  of  the  moon. 
Since  the  moon  is  smaller  than  the  sun  its  shadow  is  a 
limited  cone ;  and  the  apex  of  this  shadow  cone  sometimes 
reaches  the  earth,  when  new  moon  occurs  near  one  of  the 
lunar  nodes ;  and  sometimes  it  falls  short  of  it,  the  mean 
length  of  the  lunar  shadow  being  less  than  the  mean  dis- 
tance of  the  moon  from  the  earth.  If  the  shadow  cone 
reaches  the  earth  the  eclipse  is  total  for  all  points  within 
the  umbra ;  within  the  penumbra  the  eclipse  is  partial ;  if 
only  the  prolongation  of  the  shadow  cone  encounters  the 
earth  the  eclipse  is  annular  for  all  points  touched  succes- 
sively by  the  axis  of  the  shadow. 

169.  Speed  of  Light  from  the  Eclipse  of  Jupiter's 
Satellites  (T.,  43;  B.,  397).  — The  eclipses  of  the  inner 
satellite  of  Jupiter  occur  at  average  intervals  of  42h.  28m. 
r>iJs.  It  moves  much  faster  than  our  moon,  so  that  the 
eclipses  appear  from  the  earth  to  take  place  quite  suddenly, 


NATURE  AND   PROPAGATION  OF  LIGHT.  233 

though,  since  it  is  really  a  gradual  phenomenon,  any  one 
observation  is  doubtful  to  half  a  minute. 

In  1676  Roemer,  a  Danish  astronomer,  an  observer  at  the 
time  in  the  Paris  Observatory,  discovered  that  the  observed 
eclipses  differ  systematically  from  the  computed  times. 
When  the  earth  is  receding  from  Jupiter  the  interval  be- 
tween two  successive  eclipses  is. 
longer  than  the  mean,  and  the 
more  rapid  the  recession  the 
greater  the  excess.  The  reverse 
is  true  as  the  earth  approaches 
Jupiter.  Let  EE'  (Fig.  100) 
represent  the  earth's  orbit  and 
the  large  circle  JJ'  the  orbit  of 
Jupiter.  Then  while  the  earth 
moves  from  E  through  E'  to  E", 

or  from  opposition  to  conjunction,  the  eclipse  interval  is 
longer  than  the  mean ;  from  E"  around  to  opposition  again 
it  is  shorter  than  the  mean.  The  sum  of  all  these  excesses 
is  16m.  38s.,  or  998  seconds. 

Roemer  inferred  that  the  speed  of  light  is  finite,  so  that 
the  longer  interval  between  two  eclipses  when  the  earth  is 
receding  from  Jupiter  is  due  to  the  added  distance  which 
light  must  travel  to  reach  the  earth.  This  interval  will  be 
the  greatest  at  E'  where  the  earth  is  receding  directly  from 
the  planet.  The  sum  of  the  excesses  is  the  time  required 
by  light  to  travel  across  the  earth's  orbit.  If  the  dia- 
meter be  299,000,000  kilometres,  the  speed  of  light  will 

299,000,000 
be   -  -.1      '        —  299,600  kilometres  per  second. 

Roemer's  original  suggestion  was  rejected  by  most  as- 
tronomers for  more  than  fifty  years,  and  was  not  accepted 
till  long  after  his  death,  when  Bradley's  discovery  of  the 


234  LIGHT. 

aberration  of  light  confirmed  the  correctness  of  Roemer  s 
views  (Young's  Astronomy). 

17O.  Bradley's  Method  from  the  Aberration  of  Light 
(T.,  44;  P.,  10).  —  Aberration  is  the  displacement  in  the 
apparent  position  of  a  star  resulting  from  the  composition 
of  the  motion  "of  light  with  the  motion  of  the  earth.  It 
was  discovered  by  Bradley,  afterwards  Astronomer  Royal 
of  England,  in  1726.  If  an  observer  carrying  an  umbrella 
walk  rapidly,  while  the  rain  fails  vertically,  he  must  tilt  his 
umbrella  forward  if  he  would  protect  himself ;  for  to  him 
the  rain  does  not  appear  to  come  in  the  same  direction  as 
to  one  standing  still.  He  must  incline  his  umbrella  forward 

at  an  angle  a  with  the  vertical  so  that  tan  a  =  ^ ,  where  V 

V 

is  the  velocity  of  the  falling  drops  and  u  the  speed  with 
which  he  is  walking. 

Suppose  the  wind  blowing  directly  against  the  side  of  a 
moving  vessel,  with  a  velocity  F",  and  that  the  vessel  move 
with  a  velocity  u.  The  motion  of  the  steamer  produces 
an  apparent  wind  blowing  toward  the  stern ;  and  this  com- 
bined with  the  real  wind  causes,  to  an  observer  on  the 
vessel,  an  apparent  shifting  forward  of  the  point  from 
which  the  wind  comes  by  an  angle  a  of  which  the  tangent 

is    -  as  before.     It  was  an  observation  of  this  kind  that 

gave  Bradley  the  clue  to  the  explanation  of  the  apparent 
displacement  which  he  had  observed  in  the  position  of 
stars  when  viewed  from  opposite  sides  of  the  earth's  orbit. 
The  real  speed  of  light  must  be  combined  with  one  equal 
and  opposite  to  that  of  the  earth  in  its  orbit  in  order  to 
give  the  apparent  direction  from  which  the  light  comes, 
or  the  apparent  position  of  a  star. 


NATURE  AND    PROPAGATION  OF  LIGHT.  235 

Let  CA  (Fig.  101)  represent  the  velocity  of  light  V, 
and  let  AB  be  the  relative  magnitude  and  direction  of  the 
orbital  velocity  u  of  the  earth.  Then  will  CAD 
be  the  angle  of  aberration.  This  angle  will  be 
greatest  when  the  motion  of  the  earth  is  at 
right  angles  to  the  direction  of  the  star.  Then 

tan  a  =  —  .     The  angle  of   aberration  is  known 

to  be  almost  exactly  20".5.     Since  the  tangent 

of  this  angle  is  about  TOIJFO~}  it  follows  that  the 

speed  of  light  is  about  10,000  times  the  earth's 

orbital  velocity ;  and  as  the  latter  is  about  30  kilometres 

a  second,  the  speed  of  light  is  about  300,000  kilometres  a 

second. 

This  explanation  of  aberration  is  really  founded  on  the 
corpuscular  or  projectile  theory  of  light.  In  the  wave- 
theory  it  presents  difficulties  that  have  not  yet  been  sur- 
mounted. It  is  nevertheless  true  that  the  ratio  between 
the  speed  of  light  and  the  constant  of  aberration  is  about 

Too"  o^o* 

Both  of  these  astronomical  methods  of  measuring  the 
speed  of  light  depend  for  their  final  result  upon  an  exact 
knowledge  of  the  earth's  distance  from  the  sun.  But  it 
is  probable  that  the  most  accurate  determinations  of  this 
distance  are  to  be  made  by  reversing  the  order  of  compu- 
tations. Given  the  speed  of  light,  independently  deter- 
mined by  physical  processes,  the  constant  of  aberration,  or 
the  retardation  of  the  eclipses  of  Jupiter's  satellites,  fur- 
nishes a  measure  of  the  sun's  distance. 

171.    Pizeau's  Direct  Method  (P.,   401;   B.,  399). — 

The  first  direct  measurement  of  the  speed  of  light  over 
limited  terrestrial  distances  was  made  by  Fizeau  in  1849. 


236 


LIGHT. 


His  method  depends  upon  the  eclipse  of  a  source  of  light 
by  means  of  a  rapidly  rotating  toothed  wheel.  The  prin- 
ciple is  therefore  analogous  to  Roemer's  observations  on 
the  satellites  of  Jupiter. 

A  beam  of  light  from  a  source  8  (Fig.  102)  passes 
through  a  collimator  in  the  side  of  a  telescope ;  and,  after 
reflection  from  a  parallel  plate  of  glass  set  at  an  angle 
of  45°,  it  comes  to  a  focus  at  F,  which  is  the  principal 

focus   of    the   ob- 
E     ject   glass  of    the 
telescope.       The 


Fig.  102 


light  dive  r  ging 
from  F  emerges 
from  the  object 
glass  in  a  parallel  beam ;  and,  after  traversing  a  distance 
of  three  or  four  miles,  it  falls  upon  a  lens  Z,  which  brings  it 
to  a  focus  on  the  surface  of  the  concave  mirror  R,  having 
its  centre  of  curvature  at  the  lens  The  beam  of  light 
therefore  returns  along  its  former  path,  enters  the  telescope 
and  falls  upon  the  inclined  plate  of  glass ;  part  of  it  is  re- 
flected and  a  part  transmitted ;  and,  after  traversing  the 
eyepiece  E,  it  enters  the  eye  of  the  observer,  producing  the 
appearance  of  a  bright  star  at  F. 

A  toothed  wheel  is  so  placed  that  as  it  rotates,  the  beam 
of  light  is  alternately  intercepted  and  allowed  to  pass 
between  the  teeth.  Suppose  the  angular  breadth  of  the 
teeth  is  equal  to  the  width  of  the  spaces  between  them. 

If  now  the  speed  of  light  were  infinite,  the  illumination 
of  the  star  would  not  be  affected  by  the  rotation  of  the 
wheel ;  but  if  it  is  finite,  a  rate  of  rotation  may  be  found, 
such  that  the  light  going  out  through  a  space  will  be 
intercepted  on  its  return  from  the  distant  station  by  a 
tooth,  and  there  will  be  complete  extinction. 


NATURE  AND  PROPAGATION  OF  LIGHT.  237 

What  occurs  in  the  experiment  is  at  first  the  appearance 
of  a  bright  star,  which  gradually  diminishes  in  brightness 
as  the  speed  of  rotation  increases,  till  at  a  fixed  speed  it  is 
entirely  extinguished ;  if  the  speed  of  rotation  continues 
to  increase  the  star  reappears,  reaches  its  former  maxi- 
mum brightness,  again  fades  out,  and  is  eclipsed  when 
the  speed  of  rotation  is  three  times  that  required  for  the 
first  eclipse. 

It  was  found  very  difficult  to  decide  upon  the  exact 
speed  required  to  produce  a  complete  eclipse.  The  light 
was  much  weakened  by  successive  reflections,  so  that  the 
star  was  faint  even  at  its  greatest  brightness.  It  was 
rendered  less  distinct  by  the  diffused  light  in  the  interior 
of  the  telescope,  caused  by  reflection  from  the  teeth  of  the 
wheel.  Fizeau  found  the  first  eclipse  at  a  speed  of  12.6 
revolutions  a  second.  There  were  720  teeth,  or  1440 
divisions  of  the  wheel.  Hence  the  time  required  for  a 

tooth  to  take  the  place  of  a  space  was  ^-^  x  ^TTK  —  -i  0-144 

of  a  second.  The  double  distance  between  the  telescope 
and  the  reflector  was  17.326  kilometres.  Hence  the  speed 
of  light  as  deduced  from  this  experiment  is  17.326  x 
18144  =  314,363  kilometres. 

In  1874  Cornu  repeated  Fizeau's  experiment  with  greatly 
improved  apparatus,  the  improvements  consisting  chiefly 
in  better  methods  of  recording  the  exact  speed  of  rotation 
of  the  wheel  at  any  observed  phase  of  the  eclipse.  No 
difference  in  result  was  obtained  with  different  sources 
of  light.  Cornu's  final  result  for  the  speed  of  light  was 
300,330  kilometres  in  air.  To  obtain  the  speed  in  outer 
space,  free  from  ponderable  matter,  this  result  must  be 
multiplied  by  the  index  of  refraction  of  air  (187)  which 
gives  300,400  kilometres  per  second  in  a  vacuum. 


238  LIGHT. 

172.  Michelson's  Modification  of  Foucault's  Method 
(A.  and  B.,  434;  P.,  409;  T.,  49).  —  In  1850  Foucault 
described  a  method  of  measuring  the  speed  of  light,  founded 
upon  the  measurement  of  time  by  means  of  the  rotating 
mirror  employed  by  Wheatstone  in  1834  to  measure  the 
speed  of  electricity.  Foucault's  method  was  intended 
primarily  to  compare  the  speed  of  light  through  air  and 
water  as  a  crucial  test  between  the  emission  and  the  un- 
.  dulatory  theory  of  transmission  (188).  In  this  method  a 
I  beam  of  light  was  focused  by  a  lens  upon  a  concave  mirror 
/  after  reflection  from  a  plane  mirror.  The  centre  of  the 
^  concave  mirror  was  at  the  plane  mirror,  which  was  mounted 
to  rotate  around  an  axis  in  its  own  plane.  The  distance 
between  the  two  mirrors  was  originally  only  four  metres. 
If  the  plane  mirror  were  standing  still,  then  in  a  certain 
position  the  light  reflected  to  the  concave  mirror  would  re- 
trace its  path,  and  could  be  observed  by  means  of  an  eye- 
piece at  the  source  after  reflection  from  an  obliquely  set 
plate  of  plane  glass.  But  if  the  plane  mirror  were  rotated, 
then  the  beam  of  light,  after  travelling  to  the  concave 
mirror  and  back,  would  find  the  plane  mirror  in  a  slightly 
different  angular  position,  and  during  the  remainder  of  its 
return  passage  to  the  eyepiece  it  would  pass  by  a  slightly 
different  path.  The  divergence  of  the  two  positions  was 
measured  by  means  of  a  micrometer  in  the  eyepiece.  This 
divergence,  together  with  other  constants  of  the  apparatus 
and  the  speed  of  rotation  of  the  mirror,  gave  a  measure  of 
the  speed  of  light. 

But  in  Foucault's  arrangement  the  deflection  of  the 
beam  was  too  small  to  be  measured  with  the  requisite 
accuracy,  being  but  a  fraction  of  a  millimetre.  Michelson's 
modification  was  designed  to  increase  this  displacement.  Its 
most  important  feature  was  the  placing  of  the  lens  between 


NATURE  AND  PROPAGATION  OF  LIGHT.  239 

the  two  mirrors,  m  and  m'.  Moreover,  the  fixed  mirror, 
which  was  plane,  was  placed  at  a  distance  of  605  metres 
from  the  rotating  mirror.  The  displacement  of  the  beam 
was  increased  to  133  mm.,  or  about  200  times  that  obtained 
by  Foucault. 

An  outline  of  Michelson's  arrangement  is  shown  in  Fig. 


Fig.  103. 


103.  At  $  is  a  narrow  slit,  m  is  the  revolving  mirror,  L 
the  lens,  and  m'  the  fixed  mirror.  Light  from  a  source 
behind  S  passes  through  a  slit,  falls  on  m,  is  reflected  to 
m'  whenever  m  is  in  a  suitable  position,  and  forms  an 
image  at  S'.  Light  reflected  from  m'  through  the  lens  L 
comes  to  a  focus  at  S.  The  image  of  S  in  the  first  mirror 
and  S'  are  conjugate  foci  of  the  lens  (196). 

When  m  rotates  rapidly  its  position  will  change  while 
the  light  travels  from  m  to  m'  and  back,  and  the  reflected 
beam  will  accordingly  be  displaced  to  some  position  S"  in 
the  direction  of  the  rotation.  As  finally  arranged  the 
revolving  mirror  was  8.58  metres  from  the  slit,  and  the  dis- 
tance between  the  two  mirrors  was  605  metres.  With  a 
speed  of  257  revolutions  a  second,  the  observed  deviation 
was  113  millimetres. 

Michelson's  final  result  for  the  speed  of  light  in  a  vacuum 
was  299,853  i  50  kilometres  a  second.  By  a  similar  method, 


240  LIGHT. 

and  a  distance  between  the  mirrors  of  3720  metres,  New- 
comb  in  1882  obtained  299,860  ±  30  kilometres. 

When  a  tube  with  glass  ends  and  filled  with  water  is 
interposed  between  L  and  m'  the  displacement  of  the 
image  is  increased,  demonstrating  that  light  travels  slower 
through  water  than  through  air,  as  the  undulatory  theory 
requires. 

PROBLEMS. 

1.  A  candle  and  its  image  made  by  a  small  opening  (167)  are  at 
distances  50  and  75  cms.  respectively  from  the  opening.     Find  their 
relative  size. 

2.  If  the  intensity  of  light  varies  inversely  as  the  square  of  the 
distance  from  the  source,  find  the  relative  quantities  of  light  emitted 
by  a  gas  jet  and  a  candle  when  they  are  5  and  1.2  metres  distant  re- 
spectively from  the  photometer  disc  which  they  illuminate  equally. 

3.  Assuming  the  velocity  of  light  to   be  300,000  kilometres  a 
second,  and  the  wave-length  of  sodium  light    5890  X  10~10  metre, 
what  is  the  frequency  of  vibration  of  the  light  of  burning  sodium  ? 

4.  If  a  photograph  be  taken  in  one  ten-thousandth  of  a  second  by 
light  whose  wave-length  is   4000  X  10"10  metre,  what  is  the  length  of 
the  beam  falling  on  the  plate,  and  how  many  waves  are  impressed 
on  it? 


REFLECTION  AND   REF-R  ACTION. 


CHAPTER  IX. 

REFLECTION    AND    REFRACTION. 

173.  Law  of  Reflection  (T.,  53;  L.,  26).  —  When  a 
beam  of  light,  propagated  in  one  medium,  encounters  a 
second,  a  division  of  the  light  generally  takes  place  between 
the  two  media.  One  portion  enters  the  second  medium,  and 
follows  one  or  two  new  paths ;  the  remainder  travels  back- 
ward in  the  first  medium,  and  is  in  general  further  separated 
into  two  portions,  the  relative  intensities  of  which  depend 
upon  the  polish  of  the  surface  of  separation  between  the 
two  media.  If  this  surface  were  perfectly  polished,  all  the 
reflected  light  would  be  confined  to  a  single  direction,  and 
the  reflecting  surface  itself  would  be  invisible.  On  the 
contrary,  an  absolutely  rough,  unpolished  surface  reflects 
the  light  irregularly  in  all  directions.  This  is  ^tllaitdif 
fuseiWtg^x-  In  gen- 
eral both  processes 
go  on  together.  Non- 
luminous  objects  be- 
come visible  by  the 
diffusely  reflected 
light.  In  one  impor- 
tant class  of  cases  the 
reflection  is  total. 

When    a   beam    of 

light  falls  011  a  polished  surface  AC  (Fig.  104),  or  on  the 
surface  of   a  liquid,  a  large  part  of   it   is    reflected  in  a 


24  :>  LIGHT. 

definite  direction,  BR.     The  line   PB   is  normal  to    the 
reflecting  surface  at  the  point  of  incidence,  the  angle  IBP 
is  the  angle  of  incidence,  and  PBR  is  the  angle  of  reflection. 
The  law  of  reflection  is  as  follows  : 

The  incident  and  reflected  rays  are  in  the  same  plane 
normal  to  the  reflecting  surface,  and  make  equal  angles  with 
the  normal  at  the  point  of  incidence. 

The  two  media  considered  above  are  not  necessarily 
different  in  their  chemical  composition,  but  of  different 
optical  density.  In  special  cases  layers  of  water  or  of  air 
of  different  temperatures  give  surfaces  of  separation  at 
which  reflection  and  refraction  take  place,  as  in  the  case  of 
aerial  echoes  in  sound.  When  the  transition  in  the  density 
of  a  substance  in  the  same  molecular  state  is  gradual,  the 
reflection  is  slight,  and  the  path  of  the  refracted  ray 
becomes  curved  in  the  non-homogeneous  medium. 

It  has  been  known  from  ancient  times,  and  can  be  demon- 
strated mathematically  (V.,  II,  319),  that  if  a  ray  pass 
from  one  point  to  another,  after  reflection  from  a  fixed 
surface,  its  whole  path,  touching  the  reflecting  surface, 
is  the  shortest  that  can  be  traced  from  the  one  point  to 
the  other  when  the  angles  of  incidence  and  reflection  are 
equal  to  each  other. 

174.  Law  of  Reflection  de- 
duced from  the  Undulatory  The- 
ory (T.,  174;  P.,  66).— Let  AB 
(Fig.  105)  be  a  plane  wave-front, 
and  AC  the  reflecting  surface. 
Each  portion  of  this  surface,  as  the 
/*  wave  reaches  it,  becomes  a  new 

Fig.  105. 


centre  for  a  diverging  wave  in  the 


first  medium.  Then  during  the  time 
required  for  the  disturbance  at  B  to  reach  C,  the  disturb- 


REFLECTION  AND   REFRACTION.  243 

ance  at  A  has  spread  back  into  the  medium  as  a  spherical 
wave,  whose  radius  is  AD'  equal  to  SO.  Its  section  is  a 
circle  drawn  from  A  as  a  centre.  During  the  same,  time 
the  light  from  P,  which  would  have  proceeded  to  T1  if 
there  had  been  no  obstruction,  has  reached  §,  and  has 
developed  into  a  spherical  wave  of  radius  QT  equal  to 
QT',  whose  intersection  with  the  plane  of  the  paper  is  the 
circle  through  T.  All  the  circles  which  can  be  drawn  in 
this  manner  intersect  in  the  straight  line  CD.  This  is, 
therefore,  a  section  of  the  reflected  wave-front. 

The  triangle  ABC  equals  AD'O  equals  ADO.  There- 
fore the  angle  BAO,  which  is  the  angle  of  incidence  (135), 
is  equal  to  AOD,  the  angle  of  reflection.  Moreover,  the 
incident  and  refracted  rays  lie  in  the  plane  containing  the 
normal  line.  The  undulatory  theory,  therefore,  furnishes 
an  adequate  explanation  of  the  law  of  reflection. 

175.  Images  in  a  Plane  Mirror  (T.,  59  ;  V.,  II,  323). 
—  An  object  is  rendered  visible  by  the  rays  diverging 
from  it  and  entering  the  eye.  Hence  a  pencil  of  diverg- 
ing rays,  coming  from  a  point  or  diverging  as  if  they  came 
from  a  point,  will  convey  to  the  eye  the  impression  of  a 
luminous  source  at  that  point.  Any  perception  conveyed 
through  the  eye  is  referred  directly  outward  in  the  direc- 
tion in  which  the  light  enters  the  eye.  The  eye  can  give 
us  information  only  about  the  stimulus  which  reaches  it ; 
it  furnishes  no  direct  evidence  of  the  source  from  which 
the  stimulus  comes,  nor  of  the  manner  in  which  it  manages 
to  reach  it.  An  image  is  therefore  a  point  or  a  series  of 
points  from  which  a  diverging  pencil  of  rays  comes  or 
appears  to  come. 

From  the  general  law  of  reflection  it  results  that  all  rays 
emanating  from  the  point  A  (Fig.  106),  and  falling  upon 


244 


LIGHT, 


N 


Fig.  106. 


the  plane  mirror  MN,  have  after  reflection  the  same 
direction  as  if  they  came  from  the  point  A',  symmetri- 
cally situated  with  respect 
to  the  mirror.  In  conse- 
quence  an  eye  placed  at 
DE  will  be  affected  by 
these  rays  as  if  they  came 
directly  from  A'.  The 
point  A  is  accordingly 
called  the  image  of  A  in 
the  mirror  MN.  It  is, 
moreover,  called  a  virtual 
image  to  indicate  that  it 
is  formed  by  the  concourse 
of  the  prolongation  of  the 
rays,  and  not  by  the  rays  themselves.  An  image  does  not 
really  exist  at  A. 

(a)  The  image  of  a  point  in  a  plane  mirror  lies  on  a 
perpendicular  drawn  from  the  point  to  the  mirror,  and  as 
far  behind  the  mirror  as  the  object  is  in  front.  This 
proposition  is  readily  demonstrated  as  follows : 

Since  the  plane  of  the  incident  and  reflected  rays  con- 
tains the  normal,  the  ray  BD  projected  backward  must 
intersect  the  normal  AK  in  some  point  A.  Then  the 
angle  KAB  =  ABF  =  FBD  =  KAB.  Hence  the  two 
right  triangles,  AKB  and  A  KB,  are  equiangular ;  and 
since  the  side  KB  is  common  to  the  two,  they  are  equal 
to  each  other.  AK  is  therefore  equal  to  AK.  But 
since  ABBD  is  any  reflected  ray  from  A,  all  the  rays  after 
reflection  will  diverge  from  A',  which  is  as  far  behind  the 
reflecting  surface  as  A  is  in  front  of  it. 

(5)  The  divergence  of  the  rays  after  reflection  is  the 
same  as  before  reflection.  Hence  the  image  is  not  dis- 


REFLECTION  AND  REFRACTION, 


245 


torted.  Let  AB  and  AC  be  any  two  rays  from  A.  Then 
the  two  triangles  JJ?(7and  A' BO are  equal  to  each  other; 
for  from  the  last  pair  of  equal  triangles  AB  is  equal  to 
A'B ;  in  the  same  way  AC  \&  equal  to  A'C ;  and  since  the 
two  triangles  have  their  third  side  BC  in  common,  they  are 
equal  to  each  other,  and  the  angle  BAC  equals  the  angle 
BA'C.  But  the  former  is  the  angle  of  divergence  before 
reflection  and  the  latter  the  divergence  after  reflection. 

176.  Path  of  the  Rays  to  the  Eye.  -  -  The  image  of  an 
object  is  the  assemblage  of  the  images  of  its  points.     The 
image  may  therefore  be  found  by  dropping  perpendiculars 
from  its  several  points   upon 

the  mirror  and  producing  them 
till  their  length  is  doubled. 
Thus  A!B'  is  the  image  of  AB 
in  the  mirror  MN  (Fig.  107). 
Let  E  and  E'  represent  two 
different  observers.  To  find 
the  path  of  the  rays  entering 
the  eye  at  E,  connect  A'  and 
Bf  by  straight  lines  to  E. 
From  their  intersections  with 
the  mirror  draw  lines  to  the 
object,  A  and  B.  Then  the 
full  lines  in  front  of  the  mirror  represent  the  path  of  the 
rays  from  A  and  B,  which  give  the  image  at  A'  and  B'. 
The  rays  for  the  eye  at  E'  are  found  in  the  same  manner. 

177.  The  Deviation  by  Successive  Reflection  from 
two  Mirrors.  --  The  deviation  of  a  ray  of  light  produced 
by  two  reflections  from  a  pair  of   plane  mirrors  is  twice 
the  angle  between  the  mirrors. 


Fig    107. 


246 


LIGHT. 


Let  the  ray  be  successively  reflected  from  the  two  mir- 
rors at  E  and  F  (Fig. 
108).  Then  the  de- 
viation is  the  angle  <£. 
We  have 

---B  (/>  =  180°  —  2(e  +  i). 
0=  90°-  O  +  i). 
Doubling  the  second 

N  ^\^  equation     and    sub- 

tracting   fro  in    the 
first, 
f_20  =  6. 

But  0  is  the  angle  between  the  two  mirrors. 


Fig.  108. 


178.  Images  of  Images  (T.,  63  ;  A.  and  B.,  410  ;  V., 
II,  353).  —  When  light  is  reflected  successively  from  two 
plane  mirrors,  the  image  in  the  first  becomes  the  object  for 
the  second  mirror,  and  the  second  image  is  found  in  pre- 
cisely the  same  manner  as  the  first  one.  So  the  second 
image  may  serve  as  the  object  for  a  third  image,  and  so  on, 
since  in  each  case  the  light  approaches  either  mirror  as  if 
it  came  from  the  next  preceding  virtual  image  in  the  other 
one. 

Let  0  be  a  luminous  point  between 
the  two  mirrors  AB  and  AC  (Fig. 
109).  The  first  image  in  AB  is 
found  by  drawing  the  perpendicular 
Ob  and  making  the  distances  of  I  and 
0  from  the  mirror  equal.  Then  b  is 
in  front  of  the  mirror  AC,  and  its 
image  c'  is  determined  by  the  perpen- 
dicular be' ;  cj  is  in  front  of  the  mirror 
AB  and  has  its  image  at  b".  But  b"  is  behind  the  plane 
of  both  mirrors  and  there  is  therefore  no  image  of  it. 


Fig.  109. 


REFLECTION  AND   REFRACTION.  247 

Iii  the  same  way  may  be  found  the  images  <?,  £>',  5"  by 
first  finding  the  image  of  0  in  A  C. 

These  images  all  lie  on  a  circle  of  which  A  is  the  centre 
and  the  radius  AO.  For  the  two  right  triangles  OAB 
and  IAB,  having  the  two  sides  and  the  included  angle  of 
the  one  equal  to  the  two  sides  and  the  included  angle 
of  the  other,  are  equal.  Therefore  bA  is  equal  to  OA. 
In  the  same  way  it  may  be  shown  that  bA  equals  &A,  c*A 
equals  b"A^  etc.  All  the  images  are  therefore  equidistant 
from  A  and  lie  on  the  circle  of  which  A  is  the  centre  and 
OA  the  radius. 

When  the  mirrors  are  parallel  the  radius  is  infinite,  the 
number  of  images  is  theoretically  infinite,  and  they  are  all 
situated  on  a  straight  line  drawn  through  the  object  and 
perpendicular  to  the  mirrors.  Practically  the  number  of 
images  is  limited  by  the  rapid  decrease  in  the  intensity  of 
the  light. 

If  the  angle  6  between  the  two  mirrors  is  an  aliquot 

portion  of  four  right  angles,  or  ~  =  w,    then  when   n   is 

u 

even  the  number  of  images,  including  the  object,  is  n  ;  when 
n  is  odd  the  number  of  images  is  n  +  1.  From  the  last 
article  it  will  be  seen  that  for  every  pair  of  reflections  from 
the  two  mirrors  the  ray  suffers  a  deviation  of  twice  the 
angle  between  the  mirrors ;  and  when  it  has  changed  its 
course  by  180°  it  passes  out  between  the  two  mirrors  with- 
out further  reflection.  But  at  each  reflection  an  image  is 
formed.  Therefore  the  number  of  images  for  each  series, 
starting  first  with  one  mirror  and  then  the  other,  will  be 

2^- ,  and  the  number  for  both  series  will  be  — ,  if  6  is  con- 
tained into  TT  an  exact  number  of  times,  or  --is  an  even 

6 


248  LIGHT. 

number.  The  last  two  images  of  the  two  series  then  coin- 
cide, so  that  the  entire  number  is  w,  including  the  object. 

If  -p  is  odd,  that  is  if  ^  is  not  an  integer,  then  the  last  two 
6  u 

images  of  the  two  series  do  not  coincide  and  the  entire 
number,  inclusive  of  the  object,  is  n  +  1. 

179.    Path  of  a  Ray  from  any  Image  to  the  Eye.  - 
Suppose  the  eye  to  be  at  the  point  I  (Fig.  110)  ;  it  is  re- 
quired to  find  the  path  of  a  ray  from  the  object  to  the  eye 
for  the  third  image  I".     Draw  a  line  connecting  b"  and  I. 
It  intersects  the  mirror  AB  at  x.    Con- 
nect x  and  the  next  preceding  image 
c'  in  the  same  series  by  the  line  inter- 
secting the  mirror  A  C  in  the  point  y. 
From  y  draw  the  line  yb  to  the  next 
preceding  image,  crossing  the  mirror 
AB  at  z.     Finally  join  z  and  0.     The 
path   of   the    ray   is    Ozyxl.      If    the 
image  b'-'  is  visible,  the  light  enters  the 
eye  in  the    direction   in  which   it  is 

seen,  viz.,  b"L  But  the  light  traverses  that  path  only  up 
to  the  mirror  at  the  point  #,  which  is  the  point  of  last  re- 
flection. Now  as  I)"  is  the  image  of  c',  and  c1  acts  in  all 
respects  as  the  true  object  for  b",  the  light  must  have 
reached  the  point  x  as  if  coming  from  c'.  Therefore,  the 
line  X(/t  is  drawn.  But  this  intersects  the  other  mirror  in 
?/,  indicating  the  point  where  the  next  preceding  reflec- 
tion took  place.  The  same  process  leads  back  finally  to 
the  object. 

To  find  the  path  of  a  ray  for  any  image  from  the 
object  to  the  eye,  draw  a  line  from  the  eye  to  the  image ; 
from  its  intersection  with  the  corresponding  mirror,  draw 


REFLECTION  AND   REFRACTION.  249 

a  line  to  tlie  next  preceding  image  of  the  same  series  ; 
from  the  intersection  of  this  line  with  the  corresponding 
mirror,  draw  a  line  to  the  next  preceding  image,  and  so  on, 
till  a  line  is  drawn  from  the  last  intersection  to  the  object. 
The  portions  of  the  lines  so  drawn,  lying  in  front  of  both 
mirrors,  will  be  the  path  of  the  ray. 

If  the  first  line  drawn  does  not  intersect  the  mirror  in 
which  the  given  image  is  formed,  then  the  eye  is  not  in  a 
position  to  view  that  image. 

180.  Deviation  produced  by  the  Rotation  of  a  Plane 
Mirror.  —  If  a  plane  mirror  on  which  a  ray  of  light  falls 
be  turned  through  an  angle  about  an  axis  perpendicular  to 
the   plane  of  incidence,  the  reflected  ray  will  be  turned 
through  twice  that  angle.     Let  a  ray  of  light  AM\)Q  inci- 
dent normally  on  the  mirror  (Fig.  Ill)  ;  it  will  then  re- 
trace its   path.     If   the  mirror  is  now 

turned  through  the  angle  #,  the  normal 
is  turned  through  the  same  angle,  so 
that  the  angles  of  incidence  and  reflec- 
tion are  now  both  equal  to  6.  The 
deviation  is  then  the  angle  AMB, 
or  20. 

On  this  principle  a  plane  mirror  is 
very  extensively  used  to  indicate,  by 
the  change  in  direction  of  the  reflected  ray,  the  motion  of 
the  movable  system  of  the  instrument  to  which  the  mirror 
is  attached.  The  reflecting  galvanometer,  for  the  detection 
or  measurement  of  minute  currents  of  electricity,  takes 
its  name  from  the  mirror  which  is  attached  to  the  needle, 
and  which  indicates  the  slightest  rotational  movement. 

181.  Concave  Spherical  Mirrors  (L.,  4O ;  T.,  64). — 
Reflection  from  each  element  of  a  curved  surface  takes 


250  LIGHT. 

place  in  accordance  with  the  fundamental  law  of  reflec- 
^ionA  A  beam  of  incident  rays  gives  rise  therefore  to  a 
system  of  reflected  rays  which  can  be  geometrically  de- 
termined. 

Among  curved  mirrors  spherical  ones  are  the  simplest 
and  at  the  same  time  the  most  important. 

They  usually  take  the  form  of  a  spherical  cap,  polished 
either  on  the  interior  or  the  exterior.  The  former  are 
called  concave  ;  the  latter  convex.  The  principal  axis  is  the 
right  line  joining  the  centre  of  curvature  and  the  pole,  or 
central  point  of  the  spherical  cap. 

Let  AB  (Fig.  112)  be  a  section  of  a  concave  spherical 
mirror  through  the  principal  axis  AU.  Let  U  be  a  lumi- 

nous  point.    It  is  re- 

P/  ^ quired   to    find   the 

vl  (j^^f^  *  -TJ    formula   connecting 

r  its  distance  from  the 

mirror  with  that  of 

Fig-  112.  . 

its  image.     If  a  ray 

from  U  meet  the  mirror  at  P  it  will  be  reflected  across 
the  axis  at  V,  so  that  the  radius  OP,  which  is  the  normal 
at  the  point  of  incidence,  shall  bisect  the  angle  UP  V. 

Let  the  angle  at  U  be  represented-  by  A,  the  angles  of 
incidence  and  reflection  by  i,  the  acute  angle  at  0  by  0, 
and  the  acute  angle  at  V  by  <£.  Then  because  0  is 
external  to  the  triangle  POU  and  </>  is  external  to  the 
triangle  PVO,  we  have 

*  =  «'  +  A, (a) 


Subtracting  (6)  from  (a) 

6  —  $  =  A  -  0, 
or  20  =  A +  0. (Y) 


REFLECTION  AND   REFRACTION,  251 

If  now  P  is  very  near  A,  the  pole  of  the  mirror,  the 
angles  0,  <£,  and  A  are  very  small,  and  their  tangents  may 
be  put  equal  to  the  angles  themselves.  Let  J.77  be  repre- 
sented by  jo,  A  V  by  jt/,  and  the  radius  P  0  by  r.  Also  let 
y  equal  PA.  Then  from  (<?) 


Whence  -  +  —  .  =  -  . 

p     p'      r 

Since  y  does  not  appear  in  this  equation,  it  follows  that 
the  distance  of  V  from  the  mirror,  corresponding  to  a 
given  distance  of  U,  is  independent  of  the  position  of  the 
point  P,  to  the  extent  of  the  approximation  made  that 
the  tangent  of  an  angle  is  equal"  to  the  angle  itself.  The 
physical  interpretation  of  this  fact  is  that  for  small 
angles  of  incidence  all  of  the  rays  from  U,  incident  upon 
the  mirror,  are  reflected  so  as  to  pass  through  the  common 
point  V.  Hence  V  is  the  focus  of  the  radiant  point  Z7, 
and  the  two  are  called  conjugate  foci.  V  is  a  real  image 
because  the  rays  actually  pass  through  it  and  may  be  re- 
ceived upon  a  screen.  The  relation  between  U  and  V  is 
conjugate  because  if  V  were  the  radiant  point  the  focus 
after  reflection  would  be  U. 

182,    Principal  Focus  and  Discussion  of  the  Formula. 
-  (CL)  Since  the  sum  of  the  reciprocals  of  p  and  p'  is  a 

2 
constant  -  ,  it  follows  that  as  p  increases  p'  decreases,  and 

r 

when  p  becomes  infinite  p'  equals  •=.    Hence  parallel  rays 

22 

from  an  infinitely  distant  source  come  to  a  focus  at  a 
point  midway  between  the  centre  of  curvature  and  the 
mirror.  The  focus  for  parallel  incident  rays  is  called 
the  principal  focus. 


252 


LIGHT. 


(£)   When  p  decreases  p'  increases,  or  the  object  and 

2       2 
image  approach  each  other.     When  p  equals  p',  -  =  -  ,  or 

object  and  image  coincide  at  the  centre  of  curvature. 

•  yt 

(c)  When  p  is   less   than   r  and  greater   than  -,  p'  is 

"2t 

greater  than  r,  or  object  and  image  have  exchanged  places. 

r     1  2 

When  p  is  less  than  -~  ,  -  is  greater  than  -  ,  and  pf 


is  therefore  negative.  The  image  is  then  behind  the 
mirror  and  hence  virtual.  Distances  in  front  of  the  mir- 
ror are  considered  positive  and  those  at  the  back  nega- 
tive. When  p  is  at  the  principal  focus  the  image  is  at  an 
infinite  distance  in  either  direction.  As  p  approaches  the 
mirror  p'  also  approaches  it  from  behind,  and  the  two 
again  meet  at  the  surface  of  the  mirror. 

183.    Formation  of  Images  in  a  Concave  Mirror.  — 
Fig.  113  is  the  graphical  construction  for  the  image  when 

the  object  AB 
is  beyond  the 
centre  of  curva- 
ture C. 

For  the  pur- 
pose of  finding 
the  image  of  any 
point  of  the 

object  it  is  necessary  to  trace  the  path  of  two  rays  only,  and 
to  find  their  intersection  after  reflection.  For  convenience 
the  two  rays  selected  are  parallel  to  the  principal  axis,  and 
along  a  secondary  axis,  respectively.  A  secondary  axis  is 
any  right  line  passing  through  the  centre  of  curvature. 
The  ray  AD  after  reflection  passes  through  the  principal 


Fig    113. 


REFLECTION  AND   REFRACTION. 


253 


focus  F ;  the  ray  AC  is  reflected  directly  back  on  its  own 
path.  The  intersection  of  these  two  reflected  rays  is  at  a, 
and  a  is  therefore  the  focus  conjugate  to  A.  In  the  same 
way  the  two  rays  BE  and  BC  intersect  after  reflection  at 
6,  the  conjugate  focus  or  image  of  B.  Therefore  ab  is  the 
image  of  AB.  It  is  inverted  and  real. 

Fig.  114  is  the  con- 
struction for  a  virtual 
image  in  a  concave 
mirror.  The  ray  AD 
parallel  to  the  principal 


axis  is  reflected 
through  the  principal 
focus  F.  The  ray  CA 
retraces  its  path  after 
reflection.  The  two 

rays  themselves  do  not  meet,  but  if  the  lines  representing 
their  paths  are  produced  behind  the  mirror  they  meet  at  a. 
This  is  therefore  the  virtual  image  of  A.  Similarly  b  is 
the  virtual  image  of  B,  and  ab  is  the  image  of  AB.  It  is 
erect,  enlarged,  and  virtual. 

184.  Caustics  by  Reflection  (T.,  70;  P.,  90;  V.,  II, 
376).  —  When  the  angular 
opening  of  the  mirror  is  large 
the  approximation  made  in 
the  last  article  is  no  longer 
admissible.  Parallel  rays  do 
not  all  meet  at  the  principal 
focus  after  reflection,  and  a 
luminous  point  no  longer  has 
a  definite  image.  The  intersection;  of  the  reflected  rays 
then  form  a  luminous  surface  called  a  caustic.  A  s.ection 


254  LIGHT. 

of  this  curve  is  familiar  to  every  one  who  has  looked  at  a 
cup  of  milk  illuminated  by  a  bright  light. 

The  simplest  case  of  a  caustic  by  reflection  is  fur- 
nished by  parallel  rays  of  light  falling  on  a  concave 
spherical  mirror.  Let  AP  (Fig.  115)  be  a  section  of  the 
mirror  through  its  centre  (9,  and  let  SP  be  one  of  a  system 
of  rays  parallel  to  A  0. 

Then  since  all  the  rays  are  symmetrical  about  A  0,  if  we 
find  the  section  of  the  caustic  in  the  plane  of  the  figure, 
a  caustic  surface  will  be  generated  by  revolving  this  curve 
about  A  0  as  an  axis. 

Join  P  and  0  and  let  PQ  be  the  path  of  the  reflected 
ray.  Bisect  P  0  in  R  and  on  PR  as  a  diameter  draw  a 
circle ;  also  with  0  as  a  centre  and  with  radius  OR  con- 
struct another  circle  RB.  The  two  circles  touch  at  R. 
The  angle  QPR  equals  the  angle  ROB.  But  the  arc  QR 
subtends  QPR  at  the  circumference,  and  the  arc  BR  sub- 
tends ROB  at  the  centre  of  a  circle  of  double  radius. 
Hence  the  arcs  QR  and  RB  are  equal ;  and  if  the  small 
circle  should  roll  on  the  inner  one  the  point  Q  would  ulti- 
mately coincide  with  B,  and  would  describe  the  epicycloid 
indicated  by  the  dotted  curve.  Moreover,  the  point  of  con- 
tact R  at  any  instant  is  fixed,  and  Q  is  therefore  moving  at 
right-angles  to  QR  or  in  the  direction  of  the  reflected  ray 
PQ.  Hence  all  the  reflected  rays  touch  the  epicycloid; 
and  since  all  the  reflected  rays  are  tangent  to  the  required 
caustic,  the  epicycloid  is  therefore  a  section  of  the  caustic 
surface ;  for  the  reflected  rays  cross  everywhere  on  this 
section.  At  B  is  a  cusp  which  is  the  principal  focus  of  the 
mirror.  Not  all  parallel  rays  after  reflection  pass  through 
this  focus.  The  effect  of  this  inexactness  upon  the  image 
is  known  as  spherical  aberration* 


REFLECTION-  AND  REFRACTION.  255 

The  caustic  is  tangent  to  the  principal  axis  at  B  and  to 
the  mirror  at  E. 

This  example  furnishes  the  data  for  the  explanation  of 
the  two  focal  lines,  due  to  a  small  pencil  of  rays  incident 
at  some  point  P  ;  this  pencil  has  no  focus.  Every  reflected 
ray  PQ  passes  through  the  axis  OA  of  the  mirror.  But 
Q  is  the  intersection  of  two  successive  rays  in  the  plane  of 
the  figure.  When  the  caustic  is  made  to  rotate  about  AO 
the  point  Q  describes  a  circle  at  right  angles  to  the  plane 
of  the  paper.  Hence  a  small  pencil  of  rays  incident  about 
P  is  reflected  so  as  to  pass  through  a  short  line  at  Q,  per- 
pendicular to  the  plane  of  the  paper,  and  through  a  short 
line  along  the  axis  AO,  where  PQ  intersects  it.  Between 
these  two  mutually  perpendicular  focal  lines  is  the  circle 
of  least  confusion,  which  is  the  section  of  the  pencil  in 
which  the  rays  are  most  closely  crowded  together. 

On  account  of  the  spherical  aberration  of  concave 
spherical  mirrors,  they  cannot  be  used  for  astronomical 
purposes ;  but  if  they  have  a  parabolic  section,  all  rays 
falling  on  them  parallel  to  the  axis  will  be  reflected  so  as 
to  pass  exactly  through  the  focus. 

185.  Convex  Spherical  Mirrors.  -  -  The  formula  of 
Art.  181  is  applicable  to  a  convex  mirror.  In  this  case  the 
centre  of  curvature  and  the  radiant  point  are  on  opposite 
sides  of  the  mirror.  If  distances  in  front  of  the  mirror 
are  still  considered  positive,  r  for  this  case  is  negative. 
The  formula  then  becomes 

-+--     -- 

p      p'          r' 

All  the  images  of  a  point  in  a  convex  mirror  are  virtual ; 
for,  while  p  is  positive,  pf  must  be  negative,  since  the 
sum  of  the  reciprocals  of  p  and  p1  is  a  negative  quantity. 


256 


LIGHT. 


In  Fig.  116  the  polished  surface 


is  on  the  convex  side, 
0  is  its  centre  of  cur- 
vature, ED  is  a  ray 
parallel  to  the  princi- 
pal axis,  and  DH  is  its 
path  after  reflection. 
It  comes  as  if  from  F, 
which  is  therefore  the 
principal  focus  midway 
between  A  and  0. 
Fig.  117  shows  the 

construction  for  the  image  in   a   convex  mirror.     AD  is 

a   ray   parallel    to     the 

principal  axis,  whose  di 

rection   after    reflection 

passes  through  the  prin- 
cipal focus.   A  0  is  a  ray 

along  the  secondary  axis 

through  A.     These  two 

lines  meet  in  #,    which 

is    then    the    virtual 

image  of  A.   In  the  same 

way  the    virtual   image 

of  B  may  be  found  at 

b.     The  image  is  erect, 

smaller  than  the  object,  and  virtual. 

PROBLEMS. 

1.  An  object  6  cms.  long  is  placed  symmetrically  on  the  axis  of  a 
convex  spherical  mirror  at  a  distance  of  12  cms.  from  it ;  the  image 
formed  is  2  cms.  long.     What  is  the  focal  length  of  the  mirror  P. 

2.  A  candle  flame  is  placed  at  a  distance  of   30  cms.  from   a 
concave  mirror  made  from  a  sphere  of  30  cms.  diameter.     Find  the 
position  of  the  image.     Is  it  erect  or  inverted? 


REFLECTION  AND   REFRACTION. 


257 


3.  The  ratlins  of  a  convex  mirror  is  20  cms.  If  the  linear  dimen- 
sions of  an  object  be  twice  those  of  the  image,  where  must  each  be 
situated  ? 


186.  Refraction  (P.,  73;  V.,  II,  387;  L.,  56).  —  When 
a  luminous  ray  passes  from  one  transparent  medium  into 
another  it  undergoes  in  general  a  change  in  direction  at 
the  surface  of  separation  of  the  two  media.  The  portion 
entering  the  second  medium  is  said  to  be  refracted. 

Let  MN  (IJig.  118)  be  the  surface  of  separation.  BA 
the    incident,  and  AC  the    refracted   ray.     The  plane  of 
incidence    is    the     plane 
BAF  containing  the  inci- 
dent ray  and  the  normal 
to    MN  at   the    point   of 
incidence.      CAG-    is   the 
plane  of  refraction. 

If  the  new  medium  into 
which  the  light  is  propa- 
gated is  isotropic,  the 
plane  of  incidence  and 
the  plane  of  refraction 
coincide,  and  the  ratio  of 
the  sines  of  the  angles 
of  incidence  and  refraction  is  a  constant.  If  the  second 
medium  is  optically  denser  than  the  first,  the  refracted'  ray 
is  deflected  toward  the  normal,  as  in  the  figure.  If  the 
radius  of  the  circle  is  unity,  BF  and  CGr  represent  the 
sines  of  the  angles  of  incidence  and  refraction  respectively. 
Denoting  the  angles  of  incidence  and  refraction  by  i  and 
r  respectively,  the  law  of  sines  in  single  refraction  is 

sin  i  _ 

sin  r  ~  ^' 


Fig.   118. 


1>;~>H  LIGHT. 

The  constant  ratio  p  is  called  the  index  of  refraction. 
When  light  passes  from  a  vacuum  into  any  medium,  this 
ratio  is  called  the  absolute  index  of  refraction ;  but  when 
it  passes  from  one  medium  into  another,  it  is  called  the 
relative  index.  The  relative  index  from  medium  a  to 
medium  b  is  equal  to  the  absolute  index  of  b  divided  by 
that  of  a. 

187.  Law  of  Refraction  deduced  from  the  Undu- 
latory  Theory  (T.,  175;  P.,  74;  B.,  426).  —  Let  AB 

(Fig.  119)  be  the  trace  of  the 
incident  plane  wave,  and  A  C 
that  of  the  surface  of  separa- 
tion, both  planes  being  perpen- 
dicular to  the  plane  of  the 
paper.  Let  v  be  the  speed  of 
light  in  the  first  medium  and  v' 
that  in  the  second.  Then  if  t  is 
the  time  required  for  light  to 

traverse  the  distance  BC,  vt  equals  BO  equals  AD'.  If 
there  had  been  no  change  of  medium  the  wave-front  at  the 
end  of  the  time  t  would  have  had  the  position  D'  C,  parallel 
to  AB. 

But  the  disturbance  in  the  second  medium  travels  with 
a  speed  v'.  Therefore  a  sphere  drawn  with  a  radius  AD  — 
v't  will  limit  the  spread  of  the  wave  from  A  in  the  second 
medium.  In  the  same  way  the  disturbance  from  the  point 
P  will  travel  to  Q,  which  then  becomes  a  new  centre  of 
disturbance,  and  this  will  extend  into  the  second  medium  at 

QT       v' 

the  end  of  the  time  t  a  distance  QT,  such  that  ^—  =  _.  .  All 

(J  J.        v 

circles  representing  the  traces  of  such  spherical  waves 
ultimately  intersect  along  CD,  which  is  the  trace  of  a  plane 


REFLECTION  AND   REFRACTION.  259 

drawn  through  C  tangent  to  the  first  circle  with  radius 
AD.  CD  is  therefore  the  trace  of  the  new  wave-surface 
in  the  second  medium. 

The  angles  of  incidence  and  refraction  are  BAG  and 
A  CD  respectively.  Hence 

sin  i     sin  BAG     BO     vt      v 
sin  r     sin  A  CD      AD      v't      v' ' 

The  index  of  refraction  is  therefore  equal  to  the  ratio  of 
the  speed  of  light  in  the  first  medium  to  its  speed  in  the 
second.  The  uiidulatory  theory  thus  gives  a  satisfactory 
explanation  of  the  law  of  single  refraction.  If  the  speed  of 
light  in  a  vacuum  be  taken  as  the  unit,  the  absolute  index 
of  refraction  for  any  medium  will  be  the  reciprocal  of  the 
speed  in  that  medium. 

Michelson  obtained  for  the  ratio  of  the  speeds  in  air  and 
water  the  value  1.33,  and  for  air  and  carbon  disulphide 
1.758.  These  values  are  a  close  approximation  to  the  rel- 
ative indices  of  refraction  in  the  two  cases. 

If  the  speed  of  light  in  medium  a  is  va  and  in  medium 

b  is  vb ,  then  the  absolute  index  for  a  is  —  and  for  6,  — .  But 
the  relative  index  from  a  to  b  is  — ;  and  this  is  equal  to  the 

V 

absolute  index  of  b  divided  by  that  of  a,  or  —b. 


188.  Newtonian  Explanation  of  Refraction.  -  -  The 
Newtonian  theory  of  light  ascribes  the  change  in  the  di- 
rection which  a  ray  undergoes  at  the  surface  of  separation 
between  two  media  to  the  greater  attraction  of  the  denser 
medium  for  the  corpuscles  of  light.  The  resultant  of  all 
this  attraction  on  a  corpuscle  as  it  approaches  this  surface 


260 


LIGHT. 


must  be  along  a  normal;  therefore  the  component  of  the 
motion  parallel  to  the  surface  of  the  new  medium  will  be 
unaffected.  Let  v  and  v'  be  the  speed 
of  light  in  the  two  media,  denoted  by 
AI  and  IE  in  Fig.  120,  and  let  i  and  r 
be  the  angles  of  incidence  and  refraction. 
Then  the  component  of  v,  parallel  to  the 
surface  DC,  will  be  vs'mi,  and  of  v1  it 
will  be  v'  sin  r.  The  normal  component 
of  v  or  BI,  equal  to  IF,  will  be  increased 
by  some  quantity  FCr,  due  to  the  at- 
traction of  the  denser  matter  for  the 
corpuscle.  Therefore,  completing  the 
parallelogram  D  Gr,  its  diagonal  IE  rep- 
resents the  speed  of  light  in  the  new  medium.  It  is 
greater  than  v  for  the  rarer  medium., 

Placing  the  two  expressions  for  the  tangential  compo- 
nent equal  to  each  other, 


Whence 


E 


G 


Fig.  120. 


v  sin  i  =  v'  sin  r. 


sin 


tjj 

sin  r      v  ' 


But  by  the  undulatory  theory  the  index  of  refraction  is 
-•-,  and  v  is  greater  than  v'.  The  emission  theory,  there- 
fore, requires  that  light  travel  faster  in  the  denser  medium, 
like  water  or  glass,  than  in  the  rarer;  the  undulatory 
theory  leads  to  the  opposite  conclusion.  Foucault's  ex- 
periment showed  conclusively  that  the  emission  theory  is 
untenable,  for  light  travels  slower  in  water  than  in  air. 

189.  Refraction  at  a  Plane  Surface  (T.,  85;  V.,  II, 
4OO).  — Let  MO  (Fig.  121)  be  the  normal  to  the  plane 


REFLECTION    AND    REFRACTION.  261 

surface,  and  let  IB  be  an  emergent  ray  from  the  luminous 
point   0.     Project  the  emergent  ray  down- 
wards till    it   intersects   the    normal   at  I. 
Then  BIA  is  the  angle  of  incidence  (the 
angle  in  the  less  dense  medium)  and  BOA 
the  angle  of  refraction. 
Hence 

=  sin  BIA 
sin  BOA 

But    sin    BIA  =  sin  BIO,   since    these 
angles  are  supplementary  ;  and  as  the  sines 
of  the  angles  of  a  triangle  are  proportional  to  the  sides 
opposite, 

BO 


When  BO  is  nearly  normal,  and  the  pencil  of  rays  is 
only  slightly  divergent,  B  0  is  ultimately  equal  to  A  (9,  and 
BIto  AT.  But  /is  the  position  of  the  image  of  0.  Look- 
ing along  a  normal  line  an  object  at  0  Avill  therefore 
appear  to  be  at  I.  For  water  JJL  is  about  ^,  and  for  glass  f . 
An  object  in  water  cannot  appear  lower  than  f  its  real 
depth,  and  one  in  glass  not  further  than  f  of  its  real  dis- 
tance from  the  surface.  Conversely,  to  an  eye  under 
water  the  surface  must  appear  ^  of  its  real  distance. 

Viewed  obliquely  the  depth  of  water  appears  still  less 
than  |  of  the  actual  depth.  Hence  the  shoaling  appear- 
ance of  still  water  in  which  the  bottom  is  visible.  The 
"images  of  an  object  under  water  will  lie  on  a  caustic  sur- 
face with  the  cusp  at  f  the  real  distance  of  the  object 
beneath  the  surface.  The  cusp  will  be  on  a  line  drawn 
normal  to  the  surface  and  through  the  object. 


262 


LIGHT 


PROBLEMS. 

1.  A  straight  rod  is  partially  immersed  in  water.     The  image, 
viewed  normally,  is  inclined  45°  to  the  surface.     If  the  index  of  re- 
fraction is  f  ,  what  is  the  inclination  of  the  stick  ? 

2.  If  the  angle  of  incidence  is  60°,  and  the  index  of  refraction  is 
V  3,  find  the  angle  of  refraction. 

3.  If  the  index  of  refraction  from  air  to  water  is  ^  and  from  air  to 
glass  f  ,  what  is  the  relative  index  from  water  to  glass  ? 

19O.  Construction  for  the  Refracted  Ray.  —  Let  MN"be 
the  surface'  separating    the   two  media,   as  air  and  water 

(Fig.  122).  Let  BA  be 
incident  on  the  second 
medium  at  A.  With  A 
as  a  centre  draw  two 
concentric  circles  with 
radii  proportional  to  the 
speed  of  light  in  the  two 
media.  Their  ratio  is 
then  the  index_o_f-_re  frac- 
tion. Produce  the  line 
BA  till  it  intersects  the 
inner  circle  in  E,  and 
through  E  draw  EC  parallel  to  the  normal  DA.  It  inter- 
sects the  outer  circle  in  C.  Draw  AC  ';  it  is  the  path  of 
the  refracted  ray. 

The  triangles  EAH  and  BAF  are  similar.     Therefore 


122. 


oa  IS   AM 

But  BF  and  CGr  are  proportional  to  the  sines  of  the 
angles  BAF  and  CAG-,  and  the  angle  BAF  is  the  angle 
of  incidence.  Therefore  the  angle  CA  Cr  is  the  angle  of 
refraction. 


REFLECTION    AND    REFRACTION. 


263 


When  the  ray  passes  into  a  less  dense  medium,  through 
the  intersection  of  the  ray  produced  with  the  larger  circle 
draw  a  parallel  to  the  normal,  cutting  the -smaller  circle. 
The  line  through  this  intersection  and  the  point  of  inci- 
dence is  the  refracted  ray. 

191.  The  Critical  Angle  (T.,  86;  P.,  76). -- When 
light  passes  from  one  medium  into  another  of  smaller 
optical  density,  as  from  water  into  air,  the  ray  is  refracted 
away  from  the  normal,  and  the  angle  in  the  first  medium 
is  less  than  in  the  second.  (When  the  angle  in  the 
second  medium  becomes  a  right  angle  and  the  ray  just 
grazes  the  surface,  the  angle  in  the  first  medium  is  a  max- 
imum and  is  called  the  critical  angle\ 

Thus  the  ray  RO  in  water  ' 
(Fig.  123)  emerges  in  air  in  the 
direction  OS  ;  for  the  ray  LO  the 
corresponding  ray  in  the  rarer 
medium  is  along  the  surface  OB. 
If  the  incident  ray  in  the  denser 
medium  makes  a  greater  angle 
with  the  normal  than  LON'ik  can 
no  longer  emerge  into  the  second 
medium,  but  undergoes  total  in- 
ternal reflection.  Thus  the  ray 


10  is  totally  reflected  in  the  direction  01'.  (L  ON'  is  the 
critical  angle.)  For  a  smaller  incident  angle  part  of  the 
light  is  reflected  and  part  refracted ;  for  larger  angles  of 
incidence  it  is  all  reflected. 

To  determine  the  critical  angle  for  any  medium  whose 
relative  index  is  /A,  we  have 

sin  90° 


Sill  X 


264 


LIGHT. 


Whence 


siu  .v  =  - 


or  the  sine  of  the  critical  angle  is  the  reciprocal  of  the  index 
of  refraction. 

For  water  the  critical  angle  is 48°  27'  40" 

For  crown  glass  about 41°  10' 

For  chromate  of  lead 19°  49' 

The  total  internal  reflection  of  light  is  beautifully  shown 
by  focusing  a  beam  of  light  by  means  of  a  lens  L  (Fig. 
124)  on  the  interior  of  a  smooth  jet 
at  the  point  of  issue  0  from  the  side 
of  the  vessel.    The  angle  of  incidence 
on    the    interior   surface    of   the    jet 
then  exceeds  the  critical  angle,  and 
the   beam   is    reflected   from  side  to 
side  along  the  stream  like  sound  in 
Fig.  ,24.  a  speaking-tube.     If  colored  glass  is 

interposed  the  stream  is  colored  and 

presents  a  beautiful  appearance.  It  becomes  visible  by 
means  of  the  diffused  light  irregularly  reflected  from  fine 
matter  suspended  in  the  water. 


PROBLEMS. 

1.  In  Iceland  spar  there  are   two   refracted  rays    (225).     The 
indices  of  refraction  for  the  two  are  1.658  and  1.486.     Find  the  crit- 
ical angle  for  each. 

2.  For  chromate  of  lead  the  least  index  of  refraction  is  2.5  ;  the 
greatest,  2.95.     To  which  index  does  the  above  critical  angle  corre- 
spond ?     What  is  the  other  critical  angle  ? 

3.  What  is  the  greatest  apparent  zenith  distance  which  a  star  can 
have  as  seen  by  an  eye  under  water  ? 


REFLECTION    AND    REFRACTION. 


265 


192.  General  Construction  for  Refraction  at  a  Single 
Surface  (D.,  125).  —  Let  AB  (Fig.  125)  be  an  advancing 
wave-front  and  CD  the  bounding  surface  of  the  denser 
medium.  Through  a  number  of  points  of  AB  draw  nor- 
mals AE,  BF,  etc.  Make  all  these  normals  equal  to  one 


Fig.  125. 

another  and  connect  their  extremities  by  a  plane  curve. 
If  t  is  the  time  required  to  traverse  these  normals  with  the 
speed  v  in  the  first  medium,  then  this  plane  curve  Ea'F 
would  have  been  the  wave-front  at  the  end  of  the  interval 
t  if  no  denser  medium  had  intervened. 

To  find  the  wave-front  in  the  second  medium,  take  any 
point  d  where  the  normal  from  a  intersects  the  bounding 
surface  of  the  denser  medium  as  a  centre,  and  with  a  radius 
da",  which  bears  to  da'  the  same  ratio  that  v',  the  speed 
in  the  second  medium,  bears  to  v,  the  speed  in  the  first, 
draw  an  arc  of  a  circle.  This  defines  the  limit  to  which 
the  disturbance  from  a  has  penetrated  into  the  denser 
medium  at  the  end  of  the  time  t.  Draw  circles  in  the 
same  way  from  all  the  other  intersections  of  the  normals 
with  the  bounding  surface.  The  common  tangential  curve, 


266 


LIGHT. 


enveloping  all  the  circles  representing  the  secondary  waves, 
will  be  the  wave-front  in  the  new  medium. 

From  the  centres  of  the  secondary  waves  normals  may 
be  drawn  to  this  new  wave-front.  They  will  not  in  gen- 
eral coincide  with  aof  and  its  fellows,  but  will  show  by 
their  deviation  from  them  the  amount  of  refraction  at  each 
point  of  the  surface. 

193.  Construction  for  the  Refraction  of  a  Spherical 
Wave  at  a  Plane  Surface  (D.,  126;  P.,  83).  —Let  a 
spherical  wave  from  a  centre  0  (Fig.  126)  fall  upon  the 
plane  surface  AB  of  a  transparent  medium.  After  a  short 
interval  t  with  the  same  medium  this  wave  would  have 
taken  the  position  ADB,  a  sphere  of  radius  OD  and  centre 

0.  But  the  speed  in  the 
second  medium  is  re- 
duced along  each  normal 

in  the  ratio  of  v/  to  v  or  — •• 

t* 

Therefore  with  (7  as  a 
centre  and  a  radius  Cd 
such  that  CD  —  IJL  •  Cd,  de- 
scribe a  small  circle.  In 
the  same  way  any  other 
point  of  the  surface  AB 
'will  be  the  centre  of  a  cor- 
responding secondary  wave.  The  refracted  wave  will  be 
the  envelope  of  all  these  secondary  spherical  waves,  and  the 
new  wave-front  will  be  AdB.  The  incident  wave  is  there- 
fore flattened  down  into  another  of  smaller  curVature.  If 
tho  width  AB  be  very  small  compared  with  0(7,  the  new 
wave-front  will  be  approximately  spherical,  and  the  arc 
AdB  will  be  an  approximate  circle  with  its  centre  at  0'. 


Fig.  126. 


REFLECTION    AND    REFRACTION. 


267 


If  r  and  r'  denote  the  radii,  OD  and  O'd,  of  the  incident 
and  refracted  waves 

AC2  =  2r  •  CD  =  2r'  •  00        nearly. 
But  OD  =  p  -  Od  ; 

therefore  r'  =  /JLT, 

or  the  radius  of  the  refracted  wave  is  approximately  fi 
times  that  of  the  incident  wave.  The  true  wave-surface 
in  the  second  medium  is  hyperbolic. 

194.  Refraction  through  a  Prism  (T.,  96;  P.,  8O; 
D.,  49O).  —  A  prism  for  optical  purposes  consists  of  a 
transparent  medium  bounded  by 
two  planes  enclosing  an  angle 
which  is  less  than  twice  the  criti- 
cal angle  for  the  substance.  This 
angle  A  (Fig.  127)  is  called  the 
refracting  angle  of  the  prism. 

Since  light  is  bent  toward  the 
normal  on  entering  a  medium  of 
higher  refrangibility,  and  away  from  it  when  passing  into 
one  of  lower  refrangibility,  the  path  of  a  ray  of  homoge- 
neous light  through  the  prism  may  be  such  as  LIEO.  1 

is  the  point  of  in- 
cidence and  E  of 
emergence. 

The  precise  path 
of  the  ray  through 
the  prism  and  after 

.  i28.  emergence    may   be 

found,  when  the  in- 
dex of  refraction  is  known,  by  the  method  of  Art.  190. 
This  is  shown  applied  to  a  prism  in  Fig.  128.  Two  arcs 
are  drawn  with  the  common  centre  E,  and  radii  in  the 


Fig.  127. 


268  LIGHT. 

ratio  of  1  to  /*.  Through  the  intersection  of  the  incident 
ray  with  the  inner  arc  draw  the  dotted  line  parallel  to 
the  normal  through  E.  Through  its  intersection  with 
the  outer  arc  draw  a  line  to  E  and  produce  through  the 
prism.  This  is  EF,  the  path  through  the  prism.  In  a 
similar  way  the  two  arcs  about  F  serve  to  draw  the  emer- 
gent ray.  It  will  be  observed  that  the  construction  line 
drawn  parallel  to  any  normal  must  intersect  the  arc  of 
smaller  radius  at  the  same  point  as  the  ray  in  the  rarer 
medium,  and  the  arc  of  larger  radius  at  the  same  point  as 
the  ray  in  the  denser  medium. 

The  deviation  at  the  first  surface  of  the  prism  (Fig. 
127)  is  i  —  r  ;  at  the  second  surface,  i'  —  r1  .  Therefore  the 
total  deviation  is 

J)=i—r+i/  —  r'  =  i  +  i'  —  (r  -f  r'}. 

The  angle  of  the  prism  A  equals  the  angle  between  the 
normals  ;  and  since  this  is  the  angle  external  to  the  triangle 
IPE  at  P,  it  equals  the  sum  of  the  two  interior  opposite 
angles  r  and  r'.  Therefore 

A  =  r  +  r'. 

If  now  the  path  of  the  ray  through  the  prism  be  symmet- 
rical with  respect  to  the  two  faces  of  the  prism,  which  is 
the  condition  of  minimum  deviation  (195),  then  i  =  i'  and 
r  =  r'.  Therefore  A  =  2r  and 


Hence 


sin  i      sin  ±(A  +  D) 
and  /*=- 


sin  r  sin  i  A 


This  is  the  formula  commonly  used  for  measuring  the 
index  of  refraction  for  any  ray  whose  minimum  deviation 
is  D. 


REFLECTION    AND    REFRACTION.  269 

When  the  angle  of  the  prism  is  very  small  an  approxi- 
mate formula  may  be  found  by  making  the  angles  A  and 
D  equal  to  their  sines.  Then 


and  D=^O  —  1). 

If  a  perpendicular  be  dropped  from  A  upon  IE  when 
r  =  r',  each  half  of  the  angle  A  is  equal  to  the  angle  of 
refraction  r..  If  A  were  equal  to  or  greater  than  twice 
the  critical  angle  for  the  transparent  substance,  r  would 
be  equal  to  or  greater  than  this  critical  angle,  and  the  ray 
would  suffer  total  internal  reflection  instead  of  refraction. 
Therefore  the  angle  of  the  prism  must  be  less  than  double 
the  critical  angle. 

For  crown  glass  the  critical  angle  is 
about  41°  1(K  Hence  if  a  prism  of 
crown  glass  have  a  section  perpen- 
dicular to  its  refracting  edge  of  a  right- 
angled  isosceles  triangle  BAG  (Fig. 
129)  the  refracting  angle  will  be  more  D 
than  twice  the  critical  angle,  and  a 
ray  DE  incident  normally  on  either  Fig  |29 

face  adjacent  to  the  right  angle  will  have 
an  internal  angle  of   incidence  greater   than  the  critical 
angle,  and  will  be  totally  reflected  at  E.     For  flint  glass 
the  critical  angle  is  still  smaller. 

195.  Construction  for  Deviation  (A.  and  B.,  4O8; 
Deschanel,  1008).  —  -The  following  geometrical  construc- 
tion furnishes  a  simple  method  of  determining  the  devi- 
ation for  any  angle  of  incidence  : 

(a)    Describe  two  circular  arcs  about  a  common  centre 


270 


LIGHT. 


0  (Fig.  130),  the  ratio  of  their  radii  being  the  index  of 
refraction.     Draw  OA  for  the  direction 
of   the  incident    ray   and  through   its 
intersection   with   the   inner  arc  draw 
NA  for  the    normal.     Then  the  angle 
>B'  NA  0  is  the  angle  of  incidence  i.    Pro- 
duce NA  till  it  intersects  the  outer  arc 
in  B  and  connect  B  and  0.   Then  OBN 
is  the  angle  of  refraction  and  the  devia- 
tion is  AOB.     For 


sin  OAN     sin  i 


sin  OBN  ~~  sin  OBN 


OB 
OA 


Therefore  OBN  is  the  angle  of  refraction,  and  AOB  is  the 
difference  between  OAN  and.  OBN,  or  the  deviation. 

Draw  N'B'  making  an  angle  of  90°  with  OA,  and  -join 
B/  and  0.  Then  OB'N'  is  the  critical  angle,  for  it  is  the 
angle  of  refraction  corresponding  to  an  angle  of  incidence 
of  90°. 

(5)    To  determine  the  deviation  for  refraction  at  both 
surfaces  of  a  prism,  draw  the  circular  arcs,  the  incident 
ray,  and  the  normal  at  the  first  surface 
as  before.    Then  through  B  draw  BN' 
(Fig.  131)  to  represent  the  normal  at 
the  second  surface.     Then  OB  is  the 
direction  (not  the  path)   of  the  ray 
through  the  prism,  and  OA  is  the  di- 
rection of  the  emergent  ray.     Since 
AOB  is   the    deviation   at   the    first 
surface,  AOB  in  the  same  way  is  the  deviation   at  the 
second  surface,  and  the  total  deviation  is  the  angle  A  OA. 

(t-)  The  angle  NBN',  the  angle  between  the  normals, 
is  equal  to  the  refracting  angle  of  the  prism.  Further 


REFLECTION    AND    REFRACTION. 


271 


the  angle  A  OA'  is  measured  by  the  circular  arc  AA'.     To 

find  the  conditions  under  which  this  shall  be  a  minimum, 

we  must  ascertain  what  position  of   the  constant   angle 

ABA'  will  give  the  shortest  arc 

AA'.    Let  ABA1  and  aBa'  (Fig. 

132)  be    two    consecutive   posi- 

tions, BA  and  Ba'  being  greater 

than  BA   and  Ba.      Then    the 

arc  aaf  is  greater  than  AA  ',  both 

because   A'   is    farther  from  B 

than  A  and  because  A'a1   cuts 

across  the  angle  more  obliquely 

than  Aa;  and  A'a'  —  Aa  is  the 

increase  in  the  length  of  the  arc 

which  measures  the  deviation.     The  deviation  is  therefore 

increased  by  changing  the  position  in  such  a  way  as  to 

make  BA  and  BA'  depart  further  from  equality.     It  is 

then  a  minimum  when  BA  and  BA'  are  equal,  or  when 

the  incident  and  emergent  rays  make  equal  angles  with 

the  normals  at  the  two  surfaces  of  the  prism. 

196.  Refraction  at  Spherical  Surfaces  (T.,  1O7  ;  P., 
84).  —  Let  0  (Fig.  133)  be  the  centre  of  curvature  of  the 
spherical  surface 
A  B  of  the  denser 
medium.  Let  U 
be  the  point- 
source  of  homo- 
geneous light, 
that  is,  light  of 
one  color,  whose  index  of  refraction  is  yu.  Let  UP  be  the 
incident  ray.  Produce  it  backward  after  refraction  to 
meet  the  principal  axis  in  V. 


Fig.  133. 


272  LIGHT. 

Then  since  UPO  is  the  angle  of  incidence  and  VPO 
the  angle  of  refraction, 

sin  ;_sin  UPO  =  UU'  ^  VV  =  UU>     PV  = 
sinr     sin  VPO     PU  '  PV      VV  '  PU     **' 


„      ,     -  ,,         n  OU  ~     ,       ...      ,. 

But  from  the  figure  =  —  —  .        Substituting, 

OU    PV     OU    AV 

=-=->  when  p  1S  very  near  A 


If    we  use  the  same  notation    as    in  Art.  181,  putting 
p  for  A  U,  p'  for  A  F",  and  r  for  A  0,  then 

„=•&£. 

p'  —  r    p 
Reducing, 


If  //<  =  —  1,  this  becomes  the  formula  for  the  concave 
spherical  mirror.  For  reflection  /JL  must  have  the  value 
unity,  and  the  minus  sign  expresses  the  reversal  of  the 
ray  in  reflection. 

If  the  refracted  ray  meet  another  spherical  surface, 
very  near  the  first  one,  with  its  centre  of  curvature  on  the 
line  OA,  it  will  undergo  another  refraction.  The  dis- 
tance p'  is  the  distance  of  the  virtual  point-source  V  from 
the  second  surface  ;  and  if  we  put  q  for  the  distance  of  the 
conjugate  focus  from  the  second  surface,  and  r'  for  the 
radius  of  curvature  of  this  surface,  we  have,  since  the  re- 
fraction is  from  dense  to  rare,  and  the  index  of  refraction 

is  therefore  -  , 


/f  _  _  /* 

q      p' 

i_a« 

q      p' 


REFLECTION    AND    REFRACTION.  273 

Adding  equations  (a)  and  (6)  and 
11 

r-p  =  <*-i 

This  is  the  approximate  formula  for  a  very  thin  Zews,  a 
transparent  body  bounded  by  two  curved  surfaces,  or  one 
curved  and  one  plane.  The  distances  p  and  q  are  the  con- 
jugate focal  distances.' 

197.  The  Principal  Focus.  —  If  the  source  U  is  at  an 
infinite  distance,  p  is  infinity,  and  the  formula  becomes 


? 

If,  for  this  particular  case,  we  put  /  for  the  distance 
then 


The  distance  /  is  called  the  focal  length.  The  rays 
coming  from  an  infinite  distance  are  parallel  and  they  con- 
verge after  refraction  to  the  principal  focus. 

If  the  curvature  of  the  two  faces  is  the  same,  and  r  is 
negative, 


for  double  convex  tenses. 
If  r1  is  negative, 

1          f  1N2 

7  =  o*-i)v 

for  double  concave  lenses. 

198.  The  Sign  of  the  Quantity  /.  —  In  the  sense  in 
which  we  have  used  the  quantity  p  it  is  always  greater 
than  unity.  Therefore  the  formula  shows  that/  follows 


274 


LIGHT. 


the  sign  of  I  -        \  the  difference  of  the  curvatures  of 

the  two  surfaces.  This  focal  distance  is  positive  for  all 
lenses  thinner  at  the  centre  than  at  the  edges,  or  the  prin- 
cipal focus  lies  on  the  same  side  of  the  lens  as  the  source 
of  light ;  such  lenses  produce  a  divergence  of  the  beam. 

Distances  measured  from  the  lens  opposite  to  the  direc-- 
tion  in  which  the  light  traverses  it  are  regarded  as  posi- 
tive. Consider  the  three  forms  of  Fig.  134.  In  1,  r  is 


•-..    T 


Tig.  134. 

positive  and  r'  negative.    Therefore becomes  -  4-  ~ , 

r      T  r      r1 

a  positive  quantity.     In  2,  the  radius  r'  is  negative  and  r 

is  infinite.     Therefore is  positive.     In  3,  both  r  and 

r      r' 

r'  are  positive,  but  r  is  smaller  than  r' ;  therefore  —  ">  - 

r  ^  r' 

and  -     --is  again  positive.    All  these  lenses  are  therefore 

diverging  because  the  focal  distance  is  positive. 

For  all  lenses  thicker  at  the  centre  than  at  the  edges 

-  —  -  is  negative,   or  the  principal  focus  and  the  source 

are  on  opposite  sides  of  the  lens. 

Consider  the  three  forms  of  Fig.  135.     In  1,  r  is  negative 


and  r'  is  positive.    Therefore  —   .=. 

r      i-' 


a    nega- 
r      r1 


REFLECTION    AND    REFRACTION.  275 

tive    quantity.     In  2,  either  r  is  negative  and  r1  is  infinite, 
or  r  is  infinite  and  r'  is  positive,  according  to  the  direction 


Fig.  135. 


toward  which  the  curved  side  of  the  lens  is  turned ;  but  in 
both  cases  -    -  -^  is  negative.      In    3,    r   is    positive    and 

greater  than  r',  which  is  also  positive.     Hence  —  ]>  -  and 

-  -  is  negative.     For  all  three  forms  the  principal  focus 

and  the  source  are  on  opposite  sides  of  the  lens.     They 
are  then  all  converging  lenses. 


199.    Images  in  Lenses.  —  Since  the  quantity  (ft— 1) 

,  i  is  equal  both  to and  to  -,  these  latter  quanti- 

r      r')  q     p  f 

ties  may  be  equated,  giving 

1_1  =  1 
?     P     f 

In  the  last  article  we  have  seen  that  for  diverging  lenses 
/  is  always  positive.  Now  since  p,  the  distance  of  the 
object,  is  necessarily  positive,  q  must  be  positive  and 
smaller  than  p.  The  image  is  therefore  on  the  same  side 
of  the  lens  as  the  object,  it  is  virtual  and  nearer  the  lens 


276 


LIGHT. 


than  the  object.  Thus  in  Fig.  136,  A  is  the  source,  and  the 
ray  AB,  after  passing  through  the  lens,  diverges  from  the 
axis  as  if  it  came  from  A.  A  and  A  are  on  the  same  side 
of  the  lens,  and  A  is  nearer  the  lens  than  A. 


Fig.  136. 

For  converging  lenses /has  been  shown  to  be  negative. 
Then 

1_1=    _1 

2     P~      f 

When  -  <;  - ,  or  where  p  >/,  q  must  be  negative,  or 
object  and  image  are  on  opposite  sides  of  the  lens ;  but 
when  —  ;>  - ,  or  when  p  <^/,  q  is  positive,  or  the  image  is 

on  the  same  side  of  the  lens  with  the  object  and  is  there- 
fore virtual.     The  first  case  is  illustrated  by  Fig.  13T,  in 


Fig.  137. 


REFLECTION    AND    REFRACTION.  277 

which  the  conjugate  foci  A  and  A'  are  on  opposite  sides  of 
the  double  convex  lens.  Fig%.  138  illustrates  the  second 
case.  A  and  A  are  both  on  the  same  side  of  the  lens. 


Ffg.  138. 

For  real  images  q  is  negative.  The  formula  for  con- 
verging lenses  may  then  be  written  after  changing  all  the 
signs, 

i+i-i 

9-Pf 

When  p  =  q  we  have 

J4OT>=* 

Object  and  image  are  then  equidistant  from  the  lens, 
and  the  distance  between  them  is  4/.  They  cannot  ap- 
proach nearer  than  this  for  a  real  image. 

20O.  Image  and  Object  at  a  Fixed  Distance.  —  When 
the  object,  such  as  an  incandescent  lamp  filament,  and  the 
screen  on  which  the  image  is  received  are  at  a  fixed  dis- 
tance, which  must  be  greater  than  four  times  the  focal 
length,  there  are  two  positions  of  the  lens  which  will  give 
an  image.  In  the  first  the  lens  is  nearer  the  object  and 
the  image  is  enlarged ;  in  the  second  the  two  conjugate 
focal  distances  are  exchanged,  and  the  lens  is  nearer  the 
image,  which  is  then  smaller  than  the  object. 


278  LIGHT. 

Let  I  be  the  distance  between  the  object  and  the  screen, 
and  let  a  be  the  distance  moved  over  by  the  lens  in 
changing  the  focus  from  one  image  to  the  other. 

Then  q  +  p  =  I. 

Also  q  —  p-=a. 

Adding  these  equations  and  we  have 


2 

Subtracting  _l—a 


Therefore        ±  -  —  +  -J-  =  -*L.  , 
/      I  -\-  a     I  — a     I  —  a2 

and  hence  /  =  l*  ~  a*. 

dtt 

This  formula  furnishes  a  very  satisfactory  method  of 
measuring  the  focal  length  of  a  converging  lens.  If  the 
distance  I  be  such  as  to  make  the  two  positions  of  the  lens 
for  the  two  images  coincide,  then  a=0,  and  ?=4/,  the 
nearest  distance  of  object  and  image. 

201.  Optical  Centre  of  a  Lens.  —  Let  C  and  O'  (Fig. 
139)  be  the  centres  of  the  two  spherical  surfaces  of  the 
lens.  Draw  any  two  parallel  radii  as  AC  and  BC'.  Then 
the  tangent  planes  at  A  and  B  are  also  parallel,  and  AB 

incident  at  A  passes  through 
the  lens  as  if  through  a 
plate  with  plane  parallel 
sides.  There  is  then  no 
V  deviation  of  the  ray,  its 

path  before  incidence  being 

parallel  to  its  path  after  emergence ;  because  the  interior 
angle  of  refraction  equals  the  interior  angle  of  incidence, 


REFLECTION    AND    REFRACTION.  279 

and  so  the  external  angles  of  incidence  and  refraction  are 
also  equal.  Let  AS  be  the  path  of  such  a  ray  through  the 
lens.  It  cuts  the  axis  of  the  lens  in  0.  Then  since  AGO 
and  BOO  are  similar  triangles, 

CO  =  OA 
C'0~  OB' 

Since  the  radii  are  constant  in  value,  CO  and  OO  are 
also  constant,  the  points  C  and  O  being  fixed.  0  is  there- 
fore a  fixed  point ;  and  all  such  rays  as  AB,  whose  inci- 
dent and  emergent  portions  are  parallel,  pass  through  it. 
0  is  called  the  optical  centre  of  the  lens.  Its  distances 
from  the  two  surfaces  are  directly  as  their  radii.  For  the 
proportion  may  be  written 

00:  OB::  00:    OA. 

By  subtraction 

C'B  -  00  :  OB  ::  CA-CO:  OA, 
O'B-  OO  _  OA-CO 
0]T  OA 

But  the  numerators  are  the  distances  of  0  from  the  two 
surfaces  having  centres  O  and  C  respectively,  and  these 
distances  are  proportional  to  the  radii  of  these  surfaces. 

In  plano-convex  or  plano-concave  lenses  the  optical 
centre  is  on  the  convex  or  concave  side.  For  if  one  of  the 
radii,  as  OB,  becomes  infinite  so  that  the  corresponding 
surface  is  plane,  the  first  member  of  the  last  equation  be- 
comes zero,  the  denominator  being  infinity.  The  second 
member  is  also  zero ;  but  its  denominator  is  not  infinity ; 
therefore  its  numerator  is  zero,  or  OA  and  OO  are  equal 
to  each  other.  This  can  be  true  only  when  0  lies  on  the 
convex  surface  of  which  C  is  the  centre.  In  lenses  having 
curvatures  of  both  sides  in  the  same  direction,  the  optical 
centre  lies  without  the  lens. 


280 


LIGHT. 


PROBLEMS. 

1.  Find  the  geometrical  focus  of  a  small  pencil  of  rays  refracted 
through  a  double  CQIIVCX  lens,  the  radii  of  whose  surfaces  are  10  and 
15  cms.,  and  the  refractive  index  f,  when  the  point-source  of  the 
light  is  15  cms.  from  the  optical  centre. 

2.  The  focal  length  of  a  lens  is  20  cms.,  and  the  distance  between 
the  object  and  the  screen  100  cms.     Where  must  the  lens  be  placed 
to  give  a  clear  image  ? 

3.  The  focal  length  of  a  glass  lens  in  air  is  80  cms.    If  the  indices 
of  refraction  of  glass  and  water  are  |  and  |  respectively,  what  is  the 
focal  length  of  the  lens  in  water  ? 


202.  Construction  for  Images  in  a  Converging  Lens. 
—  (a)  When  the  object  is  farther  from  the  lens  than  the 
focal  distance.  The  image  is  then  real.  Let  AB  (Fig. 
140)  be  the  object  and  MN  &  double  convex  lens,  of  which 


Fig.   140. 


0  is  the  optical  centre.  First  draw  secondary  axes  A  0 
and  BO.  Since  these  pass  through  the  optical  centre,  the 
rays  along  them  undergo  no  deviation.  Next  draw  AD 
and  BH  parallel  to  the  principal  axis  KO,  which  passes 
through  the  centres  of  curvature  (7,  (7,  and  the  optical  centre. 
These  rays  may  be  traced  after  refraction  bfy'  the  method 
of  (190),  assuming  the  index  of  refraction  from  air  to 
glass  to  be  f .  After  refraction  at  the  second  surface  both 


REFLECTION    AND    REFRACTION. 


281 


rays  pass  through  the  principal  focus  F.  It  may  be  con- 
venient to  observe  that -if  the  index  of  refraction  is  f  and 
r  =  /•',  the  principal  focus  is  at  the  centre  of  curvature. 
Then  the  two  rays  drawn  from  A  meet  after  refraction  at 
a,  and  those  from  B  at  b.  Hence  ab  is  the  real  image  of 
AB.  Conversely,  if  ab  were  the  object  AB  would  be  the 
image.  The  image  is  inverted  because  the  secondary  axes 
cross  between  object  and  image.  The  size  of  the  image 
is  to  that  of  the  object  as  LO  to  KO. 

(b)  When  the  object  is  nearer  the  lens  than  the  focal 
distance.  The  image  is  then  virtual,  erect,  and  larger 
than  the  object.  Let  F  be  the  principal  focus  and  AB 
the  object  (Fig.  141).  Proceed  as  before  by  drawing  the 


Fig.  141 


path  of  two  rays  A  0,  B  0  along  secondary  axes,  and 
BH  parallel  to  the  principal  axis,  and  through  F  after  re- 
fraction at  both  surfaces  of  the  lens.  The  two  rays  from 
A  diverge  after  emerging  from  the  lens,  but  if  their  direc- 
tions be  projected  backwards  they  will  meet  at  a.  So  also 
the  two  corresponding  rays  from  B  emerge  from  the  lens 
as  if  they  came  from  b.  Hence  a  and  b  are  virtual  foci 
and  ab  is  the  virtual  image  of  AB.  In  this  case  object 
and  image  are  on  the  same  side  of  the  optical  centre. 


282 


LIGHT. 


203.  Construction  for  the  Image  in  a  Diverging  Lens. 
—  The  images  formed  by  concave  lenses  are  always 
virtual,  erect,  and  smaller  than  the  object.  They  may  be 
drawn  in  the  same  manner  as  those  for  converging  lenses. 

AB  (Fig.  142)  is  the  object.     The  rays  parallel  to  the 


principal  axis  after  emerging  from  the  lens  diverge, from 
the  axis  as  if  they  came  from  the  principal  focus  F.  Their 
directions  produced  backwards  intersect  the  other  two 
rays  along  the  secondary  axis  in  a  and  b  respectively. 
Hence  ah  is  the  virtual  image  of  AS. 

2O4.  Spherical  Aberration  and  Distortion  of  Image 
(B.,  444;  P.,  85).  — It  must  not  be  overlooked  that  the 
formula  which  we  have  used  to  connect  the  object  and 
the  image  with  the  focal  length  of  a  lens  is  only  an 
approximate  one.  It  was  deduced  by  assuming  the  point 
of  incidence  on  the  lens  very  near  the  principal  axis,  and 
the  point-source  of  light  was  supposed  to  be  on  the  axis. 

If  the  approximations  made  in  deducing  the  formula 
(196)  are  carefully  examined,  it  will  be  found  that  the 
focal  length  for  rays  incident  on  a  lens  near  the  boundaries 
of  the  spherical  surfaces  is  shorter  than  for  rays  incident 
near  the  centre.  It  follows  that  when  parallel  rays  are 


REFLECTION    AND    INFRACTION.  283 

incident  on  a  lens  the  focus  for  the  peripheral  rays  is 
nearer  the  lens  than  for  the  central  rays.  This  distribu- 
tion of  the  focus  along  the  axis  is  called  spherical  aber- 
ration. Between  the  two  foci  for  the  outer  and  the  inner 
rays  the  refracted  beam  will  have  a  minimum  cross-section, 
which  is  called  the  circle  of  least  confusion.  The  rays  re- 
fracted in  this  manner  from  different  parts  of  the  lens  are 
tangent  to.  a  curve  called  a  caustic  by  refraction. 

The  corrections  for  spherical  aberration  are  two  :  First, 
the  outer  rays  are  cut  off'  by  an  annular  diaphragm.  This 
is  partly  the  office  of  the  iris  of  the  eye.  Second,  the 
surfaces  of  the  lens  are  so  shaped  that  a  spherical  wave 
before  refraction  remains  spherical  after  refraction.  The 
surfaces  of  the  lens  are  then  only  approximately  spherical. 

A  surface  which  refracts  to  one  point  the  light  diverg- 
ing from  another  is  called  an  aplanatic  surface. 

The  formulae  and  discussions  to  which  this  inquiry  leads 
are  tedious  and  intricate,  and  do  not  come  within  the 
limits  prescribed  for  this  book. 

^Besides  the  indistinctness  or  confusion  of  the  image  to 
which  spherical  aberration  leads,  there  is  a  distortion  with 
spherical  lens  surfaces.  It  has  been  assumed  that  a  straight 
object  will  have  a  straight  image.  But  with  spherical 
lenses  the  image  of  a  straight  line  at  one  side  of  .the  axis 
is  a  curved  line,  and  the  image  of  a  plane  surface  is 
convex.!  From  the  equation 

1    _1        1 
?      P       f 

we  obtain 


~—  . 
p     J  +  p 

Now  q  is  the  distance  of  the  object  and  p  that  of  the 
image.     Hence,  while  q  increases  with  p,  both  measured 


284  ^  LIGHT. 

along  secondary  axes,  for  points  of  a  straight  object  further 

and  further  from  the  principal  axis,  yet  it  does  not  increase 

^ 
in  the  same  ratio,  since  — - —  diminishes   as   p  increases. 

f+P 

This  means  that  the  image  ab  (Fig.  141),  for  example,  is 
concave  toward  the  lens,  the  distance  of  its  outer  parts,  a 
and  5,  from  0  bearing  a  smaller  ratio  to  the  corresponding 
distances  of  A  and  B  than  those  of  points  near  the  axis 
to  the  corresponding  points  of  the  object. 


DISPER  SION.  28  5 


CHAPTER   X. 

DISPERSION. 

205.  The  Complexity  of  White  Light  (V.,  II,  475). 
—  When  a  thin  beam  of  sunlight  is  passed  through  a 
prism  it  not  only  suffers  deviation,  as  already  explained, 
but  it  is  resolved  into  a  number  of  colors  of  different  re- 
frangibility.  This  phenomenon  is  known  as  dispersion. 

The  production  of  colors  in  this  way  was  certainly  known 
to  Seneca,  and  Kepler  made  use  of  an  equilateral  glass 
prism  for  the  study  of  the  subject.  But  Newton  was  the 
first  to  recognize  the  true  import  of  the  phenomenon,  and 
to  refer  the  colors  to  the  heterogeneity-  of  white  light. 
This  was  in  1666. 

Admit  a  horizontal  beam  of  sunlight  into  a  darkened 
room  through  a  narrow  vertical  slit  and  focus  by  means  of 
a  long-focus  lens  on  a  screen  at  a  suitable  distance.  Then 
interpose  in  the  path  of  the  beam  a  glass  prism  (or,  better, 
one  with  plane  glass  faces  filled  with  carbon  disulphide) 
with  its  refracting  edge  vertical.  The  beam  of  light  will 
undergo  deviation  and  dispersion,  and  the  screen  should 
be  moved  to  receive  it,  keeping  its  distance  from  the  lens 
the  same.  The  order  of  colors  is  red,  orange,  yellow, 
green,  blue,  violet.  Another  color,  indigo,  is  sometimes 
distinguished  between  blue  and  violet.  Red  is  the  least 
refrangible,  and  violet  the  most.  Each  color  has  a  differ- 
ent refrangibility.  The  color  is  the  physiological  character 
of  a  light ;  the  refrangibility  is  its  physical  character. 


•JSC)  LIGHT, 

Such  a  succession  of  colors  in  the  order  of  refrangibility, 
obtained  from  any  source  of  light,  is  called  a  spectrum. 

The  index  of  refraction  for  red  is  less  than  for  violet ; 
and  since  the  relative  index  is  inversely  as  the  speed  of 
light  in  the  medium,  it  follows  that  red  light  is  transmit- 
ted through  the  prism  with  a  greater  speed  than  violet ;  the 
other  colors  are  transmitted  through  transparent  media 
with  intermediate  speeds.  The  phenomenon  of  dispersion 
is  therefore  due  to  the  unequal  retardation  in  the  speed  of 
transmission  of  the  different  colors  through  transparent 
media.  Violet  suffers  a  greater  retardation  or  travels  more 
slowly  than  red  on  entering  an  optically  denser  medium. 
Measurements  of  wave-length  show  that  the  ethereal  un- 
dulations producing  extreme  violet  are  the  shortest  of  all 
those  coming  within  the  visible  spectrum.  It  follows  that 
disturbances  of  short  wave-length  undergo  greater  dim- 
inution of  speed  on  entering  dense  transparent  media  than 
those  of  longer  period. 

There  is  no  evidence  that  ethereal  undulations  of  differ- 
ent wave-length  travel  with  different  speeds  in  the  ether 
of  space. 

2O6.  Recomposition  of  White  Light.  -  -  Newton's 
conclusion  that  refraction  does  not  produce  the  colors,  but 
serves  only  to  separate  those  already  mingled  in  white 
light,  was  confirmed  by  recombining  the  separated  colors 
into  a  beam  of  white  light.  One  of  the  methods  employed 
for  this  purpose  was  the  reflection  to  one  point  by  seven 
mirrors  of  the  different  rays  of  the  spectrum.  The  super- 
position of  these  produced  white  light.  But  a  more  con- 
clusive experiment  was  the  actual  synthesis  of  white  light 
from  the  spectral  colors  by  means  of  a  second  prism  iden- 
tical with  the  first  and  placed  with  its  refracting  edge 


1 


287 

turned  in  the  opposite  direction  (Fig.  143),  where  S  is 
the  incident  beam  which  is  re- 
solved into  the  spectral  colors 
by  the  first  prism  and  recom- 
bined  into  an  emergent  beam  E 
of  white  light. 

207.  Dispersive  Power.  —  If  two  prisms  of  different 
materials  are  made  with  such  angles  that  they  give  the 
same  minimum  deviation  for  the  brightest  part  of  the  spec- 
trum, it  will  be  found  that  the  lengths  of  the  two  spectra 
will  not  be  the  same.  The  angular  separation  of  the 
colors  varies  with  the  transparent  medium  employed.  If 
6?',  d",  d,  represent  the  two  extreme  and  the  mean  devia- 
tions in  the  spectrum  and  ///,  ft",  ft,  the  corresponding 
indices  of  refraction,  then  d'  —  d"  is  the  angular  separation 
of  the  two  extreme  colors  of  the  spectrum,  or  the  disper* 
sion.  But  by  Art.  194,  when  the  refracting  angle  is 
small,  the  deviation  is  A  (ft  —  1).  Therefore 

d'  —  d"_A  (ft7  —  1)  —  A  (ft"— 1)  _  f/  —  ft"  _  dp 
d  A(/JL  —  1)  /*  — 1       ft  — 1' 

and  this  ratio  is  called  the  dispersive  power  of  the  sub- 
stance. It  is  the  ratio  of  the  difference  of  deviations  of 
two  selected  rays  of  the  spectrum  to  the  mean  deviation. 
This  ratio  is  constant  for  the  same  substance  so  long  as 
the  refracting  angle  of  the  prism  is  small,  but  it  is  different 
for  different  substances.  Thus  for  crown  glass  the  disper- 
sive power  is  0.0434,  while  for  carbon  disulphide  it  is 
0.1406.  For  the  same  deviation,  therefore,  a  hollow  prism 
filled  with  carbon  disulphide  will  give  a  spectrum  more 
than  three  times  as  long  as  the  one  produced  by  a  prism 
of  crown  glass. 


288  LIGHT. 

208.    Chromatic  Aberration  (T.,  113;  VM  II,  568).  — 

Since  the  homogeneous  colors  of  white  light  have  different 
indices  of  refraction,  it  follows  from  the  formula 


that  a  single  lens  has  different  focal  lengths  for  different 
colors,  and  that  /  is  less  as  ^  is  greater.  Hence  violet 
light  comes  to  a  focus  nearer  the  lens  than  red.  Thus  in 

Fig.  144  v  is  the  principal  focus 
for  violet  rays  and  r  for  red.  The 
other  colors  have  foci  lying  be- 
tween these  two.  If,  therefore,  a 
screen  be  placed  at  or  near  v,  as 
at  #,  the  image  will  be  bordered 

with  red ;  if  at  y,  near  the  focus  r,  it  will  be  fringed  with 
violet.  The  image  with  least  color  will  be  obtained  by 
placing  the  screen  midway  between  the  two  foci  where 
the  beam  of  light  has  its  smallest  cross-section ;  but  a 
colorless  image  cannot  be  obtained  with  a  single  lens. 
This  confusion  of  colored  images  is  called  chromatic  aber- 
ration. 

Newton  believed  that  these  colors  were  unavoidable. 
According  to  him,  the  dispersion  being  proportional  to  the 
refraction,  the  one  could  not  be  suppressed  without 
annulling  the  other.  But  Euler  observed  that  the  eye 
gives  colorless  images  ;  and  as  this  organ  is  composed  of 
several  refracting  media,  he  concluded  that  one  should 
be  able  to  make  achromatic  lenses  by  means  of  two  or 
more  glass  lenses  .united  by  some  liquid. 

Newton's  experiments  were  shown  to  be  inexact,  and  the 
English  optician  Dolland  succeeded,  by  the  use  of  a  prism 
of  glass  and  one  of  water  with  a  variable  angle,  in  making 
the  colors  disappear  while  retaining  still  a  certain  amount 


DISPERSION.  289 

of  deviation.  A  little  later  (1757)  he  succeeded  in 
making  an  achromatic  lens  by  a  combination  of  crown 
and  flint  glass. 

2O9.  Conditions  of  Achromatism  (T.,  114;  V.,  II, 
569 ;  D.,  492  ;  B.,  452).  —  It  will  be  evident  from  a  con- 
sideration of  Art.  207  that  by  varying  the  refracting  angle 
of  two  thin  prisms  of  different  materials,  such  as  crown 
and  flint  glass,  and  by  combining  them  with  their  sharp 
edges  turned  in  opposite  directions,  it  is  possible  to  secure 
deviation  without  dispersion,  or  dispersion  without  deviation. 
The  conditions  governing  the  first  are  those  required  to 
secure  an  achromatic  image  with  two  lenses ;  those  of  the 
second  apply  to  a  direct-vision  spectroscope. 

The  expression  for  dispersion  is  AAp.  Suppose  we  have 
a  second  prism  for  which  the  dispersion  is  A'J/ji'.  Then  if 
the  image  is  to  be  colorless  the  dispersion  of  the  colors  in 
one  direction  must  equal  the  dispersion  in  the  other ;  that 
is,  the  angles  of  the  prisms  must  be  such  that  the  disper- 
sion shall  be  the  same  for  both.  Then 
A  dp  -  A'Jp'  =  0, 

£=^:- 

A'      Ap 

•iHence  the  condition  of  achromatism  for  the  two  prisms 
is  that  their  refracting  angles  must  be  inversely  propor- 
tional to  the  differences  between  the  indices  of  refraction 
of  the  two  pairs  of  selected  rays  near  the  limits  of  the 
spectrum.^' 

But  since  these  two  prisms,  which  are  assumed  to  have 
different  dispersive  powers,  produce  the  same  dispersion,  the 
deviations  produced  by  them  cannot  be  the  same. 


290  LIGHT. 

and  the  numerators  are  equal,  it  follows  that  the  devia- 
tions, represented  by  the  denominators,  are  unequal. 
There  will,  therefore,  be  deviation  without  dispersion,  and 
the  deviation  will  be  due  to  the  material  having  the 
smaller  dispersive  power. 

But  an  achromatic  combination  of  two  lenses,  as  well  as 
two  prisms,  may  be  used  to  superpose  the  foci  for  two 
given  colors.  Suppose  the  system  composed  of  two  very 
thin  lenses  placed  in  contact.  The  conditions  of  approxi- 
mate achromatism  are  not  difficult  to  find.  The  power  of 
a  lens  is  the  reciprocal  of  its  focal  length,  and  the  re- 
fracting power  of  a  system  of  two  lenses  of  vocal  length 
/  and/',  and  very  near  together,  is  given  by  the  equation 

1  =  1  +  ! 

F     f      f  ' 

or          l- 


where  //.  and  i*f  are  the  indices  of  refraction  of  the  two 
kinds  of  glass  composing  the  two  lenses  for  a  definite  ray 
of  one  color,  as,  for  example,  red.  The  r's  are  the  radii  of 
curvature  of  the  four  surfaces. 

For  the  other  simple  light,  violet,  for  example,  the  in- 
dices for  the  tv/o  lenses  being  yn  +  Ap  and  p'  +  J/x',  the 
focal  distance  of  the  system  becomes  F—  AF.  Now  in 
order  that  the  foci  of  the  two  selected  colors  may  coincide 
AF  must  be  zero.  Obviously  this  will  be  the  case  when 


since  these  are  the  only  terms  entering  into  the  expression 

for  F  —  AF  which  differ  from  the  expression  above  for  Ft 

Since  Ap  and  Ap1  are  of  the  same  sign,  the  parenthetical 

quantities  must  be  of  opposite  signs;  that  is,  if  one  of  the 


DISPERSION.  291 

lenses  is  convergent,  the  other  is  divergent.     The  system 
as  a  whole  will  be  convergent  if  the  converging  lens  is 
composed  of  glass  of  the  smaller  dispersive  power. 
From  the  equation  for/,  viz., 


we  have   -— v-  —         7  •  -  • has  a  similar  value. 

T!     r2      ^  — 1    /'    r3      r4 

Substituting  in  the  equation  of  condition  above  and 


\ 


JFrom  this  equation  we  see  that  in  an  achromatic  system 
of  two  lenses  their  focal  lengths  are  proportional  to  their 
dispersive  powers.^ 

The  above  equations  may  be  used  to  determine  the  four 
radii.    Generally  rsis  made  equal  to  —  r2  and  the  other  two 
are  arranged  to  make  the  spherical  aberration  as  small  as 
possible.     Fig.  145  shows  a  combination  of 
flint    and   crown    glass    for    achromatism. 
Since  the  dispersive  power  of  crown  glass 
is  only  about  half  that  of  flint  (0.0434  and 

Fig.  145. 

0.0753)   the    converging   lens  is  of   crown 
glass  and  the  diverging  of  flint  for  a  converging  combina- 
tion.    The  outer  surface  of  the  flint  glass  is  not  usually 
plane,  nor  are  the  radii  of  curvature  of  the  two  surfaces  of 
the  crown  glass  equal  to  each  other  for  the  best  definition, 

PROBLEMS. 

REFRACTIVE  INDICES  FOR  THE  FRAUNHOFER  LINES. 

ABCDEFGII 

Crown  Glass  .  .  1.5089  1.5109  1.5119  1.5146  1.5180  1.5210  1.5266  1.5314 
Flint  Glass  .  .  .  1.6391  1.6429  1.6449  1.6504  1.6576  1.6642  1.6770  1.6886 
Water  ....  1.3284  1.3300  1.3307  1.3324  1.3347  1.3366  1.3402  1.3431 
Carbon  Bisulphide,  1.6142  1.6207  1.6240  1.6333  1.6465  1.6584  1.6836  1.7090 


292  LIGHT. 

1.  Find  the  dispersive  power  of  crown  glass  and  flint  glass  for 
the  lines  A,  H,  and  D. 

2.  If  the  refracting  angle  of  a  crown-glass  prism  is  20°,  what 
must  be  that  of  a  flint-glass  prism   to    produce  achromatism,  and 
what  will  be  the  resulting  deviation  for  A,  H,  and  E? 

3.  Suppose  we  wish  to  reunite  the  colors  represented  by  the  7> 
and  O  lines  in  crown  and  flint  glass ;  what  must  be   the   relative 
focal  lengths  of  the  two  lenses  of  the  combination  for  the  E  line  ? 

21O.  Irrationality  of  Dispersion  (B.,  452  ;  P.,  191 ;  D., 

493) If  the  dispersion  or  angular  separation  of  two 

colors  be  made  the  same  for  prisms  of  any  two  media,  the 
dispersion  for  the  other  colors  of  the  spectrum  will  not  in 
general  be  the  same.  While  the  order  of  the  colors  in  the 
two  spectra  is  the  same,  except  for  a  few  abnormal  cases, 
yet  the  relative  spaces  occupied  by  the  several  colors  are 
not  the  same.  This  is  known  as  the  irrationality  of  dis- 
persion. Thus  the  spectra  produced  by  prisms  of  crown 
and  flint  glass  may  be  made  of  the  same  length  by  prop- 
erly adjusting  the  refracting  angles,  but  the  lengths  occu- 
pied by  the  same  colors  in  the  two  will  not  be  the  same. 
For  crown  glass  orange  and  yellow  are  spread  over  a  rela- 
tively greater  area  than  for  flint  glass.  The  relative  dis- 
tribution of  the  colors  throughout  the  refraction  spectra 
of  different  transparent  media  is  not  the  same. 

Obviously,  therefore,  the  achromatism  secured  by  com- 
bining two  lenses  must  be  of  an  imperfect  kind.  Cor- 
rection for  the  extreme  colors  does  not  correct  for  the 
intermediate  ones;  and  there  remains,  therefore,  a  resid- 
uum of  color.  Only  two  colors  can  be  superposed  by  a 
double  combination  of  lenses.  To  superpose  three  colors 
a  triple  combination  is  necessary.  For  a  double  combina- 
tion rays  of  extreme  refrangibility  are  not  generally 
chosen,  but  the  orange-yellow  and  the  greenish-blue. 


DISPERSION. 


293 


211.  Dark  Lines  in  the  Solar  Spectrum.  —  A  spectrum 
consists  of  a  s accession  of  colored  images  of  the  slit.  If 
this  slit  is  not  extremely  narrow  the  several  images  will 
overlap,  giving  what  is  called  an  impure  spectrum.  To 
obtain  a  pure  spectrum  the  slit  must  be  narrow  aiid  an 
achromatic  lens  must  be  placed  at  a  distance  from  the  slit 
equal  to  its  focal  length,  so  that  the  light  emerging  from 
it  may  consist  of  parallel  rays.  These  parallel  rays  may 
be  received  upon  a  prism  adjusted  for  minimum  deviation. 
The  resulting  pure  spectrum  should  then  be  viewed  with 
an  achromatic  telescope.  A  piece  of  apparatus  for  obtain- 
ing and  viewing  a  pure  spectrum  is  called  a  spectroscope. 
When  it  is  provided  with  a  divided  circle  for  measuring 
deviations  it  is  called  a  spectrometer.  A  simple  form  of 
spectrometer  is  shown  in  Fig.  146.  At  the  left  is  an  ad- 
justable slit  placed  at  the  principal  focus  of  the  lens 
contained  in  the 
other  end  of  the 
same  tube.  Paral- 
lel rays  pass  from 
this  tube  to  the 
prism  and  are  there 
refracted  so  as  to 
pass  into  the  ob- 
serving telescope 
on  the  right.  This 
is  movable  so  as  to  take  successive  parts  of  the  spectrum 
into  the  field  of  view. 

When  the  solar  spectrum  is  examined  by  such  a  spec- 
troscope it  is  found  to  be  crossed  by  numerous  dark  lines. 
These  dark  lines  were  first  observed  by  Wollaston  in  1802 
by  looking  at  a  slit  in  the  shutter  of  a  dark  room  through 
a  prism  held  in  the  hand  at  a  distance,  and  with  its 


Fig.  146. 


294  LIGHT. 

refracting  edge  parallel  to  the  slit.  The  rays  reaching 
the  prism  were  then  nearly  parallel.  Fraunhofer  studied 
them  in  1814-15,  counted  about  600  of  them,  and  marked 
the  places  of  354  on  a  map  of  the  spectrum.  They  are 
therefore  often  called  Fraunhofer's  lines.  The  solar 
spectrum  is  discontinuous,  or  is  characterized  by  the  ab- 
sence of  numerous  colored  images  of  the  slit.  These  dark 
lines  represent  the  places  of  rays  of  definite  refrangibility 
or  wave-length.  Some  of  them  are  always  present  in  the 
solar  spectrum.  These  have  their  origin  in  the  sun  itself. 
Others  are  greatly  strengthened  or  appear  only  as  the 
sun  nears  the  horizon.  These  variable  lines  have  their 
origin  in  the  earth's  atmosphere,  and  are  called  atmos- 
pheric or  telluric  lines. 

Fraunhofer  designated  the  chief  lines  mapped  by  him 
A,  B,  0,  D,  E,  F,  a,  H.  He  afterwards  added  the  line  a 
in  the  red  and  the  line  b  in  the  green.  These  dark  lines 
extend  from  A  in  the  extreme  red  to  TI  in  the  extreme 
violet. 

212.  Three  Kinds  of  Spectra.  --  When  the  spectra 
from  different  sources  of  light  are  classified  they  fall  into 
three  groups,  each  of  which  may  have  several  subdivisions : 

(a)  Bright-line  spectra.  The  spectra  of  incandescent 
gases  and  vapors  consist  of  a  limited  number  of  br!</Jtt 
lines,  each  of  which  is  a  monochromatic  image  of  the  slit 
—  an  image  in  one  color.  Thus  the  spectrum  of  sodium 
vapor  is  the  yellow  line  marked  D  on  the  maps.  With  a 
spectroscope  of  sufficient  resolving  power  this  yellow  line 
is  found  to  consist  of  two  lines,  and  each  one  of  these  is 
double.  Other  vapors  produce  other  colored  lines,  and  no 
two  vapors  or  gases  give  rise  to  the  same  series  of  bright- 
line  images  of  the  slit. 


DISPEBSION.  295 

(£>)  Continuous  spectra.  Next  in  the  order  of  complexity 
come  the  continuous  spectra  of  incandescent  or  white-hot 
solids  and  liquids.  They  exhibit  a  perfectly  continuous 
succession  of  colors  from  one  extremity  of  the  spectrum 
to  the  other  without  any  interruptions  or  gaps.  The 
spectra  of  lights  used  for  artificial  illumination,  such  as 
those  of  candles,  gas  flames,  or  the  electric  light,  are  con- 
tinuous. Their  light  is  due  chiefly  to  white-hot  carbon, 
an  incandescent  solid.  The  extension  of  these  spectra 
toward  the  more  refrangible  or  violet  end  depends  upon 
the  temperature  of  the  source. 

(c)  Absorption  spectra.  These  are  discontinuous,  like 
the  solar  spectrum,  and  they  are  rendered  such  by  losses 
due  to  absorption  in  the  passage  of  light  through  trans- 
parent media.  The  absorption  producing  the  dark  lines  of 
the  solar  spectrum  takes  place  chiefly  in  the  outer  envelope 
of  the  solar  atmosphere.  The  incandescent  mass  of  the 
sun,  which  would*  by  itself  present  a  continuous  spectrum, 
is  surrounded  by  an  atmosphere  of  gases  and  vapors  at  a 
high  temperature,  traversing  which  are  the  rays  emitted  by 
the  photosphere.  Absorption  takes  place  in  this  reversing 
layer;  and  the  dark  rays  produced,  which  are  only  rela- 
tively dark  in  comparison  with  the  adjacent  bright  portions 
of  the  spectrum,  are  the  inversion  of  those  luminous  rays 
which  form  the  emission  spectra  of  these  gases  and  vapors 
in  the  outer  envelope  of  the  sun. 

The  principle  of  absorption  is  the  same  as  that  of  reso- 
nance or  co-vibration  in  sound.  Every  gas  or  vapor  when 
white  hot  emits  rays  of  the  same  wave-length  as  those 
which  it  absorbs  from  an  independent  source  when  at  a 
lower  temperature.  Thus  the  D  line  of  the  solar  spectrum 
coincides  exactly  with  the  bright  line  given  by  sodium 
vapor  in  a  state  of  incandescence.  Not  only  has  the 


296  LIGHT. 

coincidence  been  established  between  the  Fraunhofer  D 
line  and  the  yellow  line  of  sodium  vapor,  but  the  reversal 
of  this  yellow  line  by  sodium  vapor  as  the  absorbing  agent 
has  been  accomplished.  These  results  laid  the  foundation 
for  the  science  of  spectrum  analysis  by  which  the  approxi- 
mate chemical  composition  of  self-luminous  celestial  bodies 
has  been  made  out. 

Kirchhoff  in  1860  established  the  following  law  of  spec- 
trum analysis : 

The  relation  between  the  emissive  power  and  the  absorbing 
power,  relative  to  any  definite  radiation,  is  the  same  for  all 
bodies  at  the  same  temperature.  If  light  from  a  vapor  at  a 
higher  temperature  traverses  the  same  vapor  at  a  lower 
temperature,  the  light  absorbed  in  the  latter  is  greater  than 
the  light  emitted  by  it.  The  result  is  relatively  dark  lines 
or  a  reversal. 


INTERFERENCE    AND    DIFFRACTION.  297 


CHAPTER   XI. 

INTERFERENCE    AND    DIFFRACTION. 

213.  Interference  of  Light  from  two  Similar  Sources 
(T,,  185  ;  B.,  488;  P.,  119).  --  The  phenomena  resulting 
from  the  superposition  of  two  systems  of  waves  of  homo- 
geneous light,  travelling  in  nearly  the  same  direction,  are 
called  interference.  Similar  phenomena  have  already  been 
discussed  in  sound. 

When  the  ether  at  any  point  is  affected  simultaneously 
by  two  waves  it  is  thrown  into  vibration  by  both,  and  the 
result  is  a  compound  motion  into  which  each  vibration 
enters  independently.  If  the  two  vibrations  are  in  the 
same  direction,  their  amplitudes  are  added  and  the  result- 
ing amplitude  is  the  sum  of  the  two  components ;  but  if 
they  are  of  opposite  sign  the  resultant  amplitude  is  equal 
to  the  difference  of  the  constituent  amplitudes,  If  in  this 
latter  case  the  amplitude 3  of  the  two  vibrations  are  equal 
the  two  motions  should  completely  annul  each  other. 

We  know  already  that  two  systems  of  sound-waves  may 
interfere  so  as  to  produce  silence,  and  two  water-waves 
may  be  superposed  so  as  to  leave  the  surface  undisturbed. 
If  the  undulatory  theory  of  light  is  true  it  should  be  pos- 
sible to  add  light  to  light  in  such  a  manner  as  to  produce 
darkness.  This  actually  occurs  in  many  cases.  The  limits 
set  to  this  book  will  enable  us  to  explain  only  a  few  of  the 
most  simple  ones. 

The  most  simple  arrangement  to  exhibit  interference 


298  LIGHT. 

was  devised  by  Fresnel.  BCD  (Fig.  147)  is  an  isosceles 
glass  prism  with  the  angle  at  C  nearly  180°.  Let  0  be  a 
source  olMlbmogeneous  light  in  a  plane  through  the  angle  0 
of  the  prism  perpendicular  to  BD.  The  light  passing 


through  the  prism  will  consist  of  two  parts  diverging  from 
the  virtual  sources  Ol  and  02.  Since  the  point  A  is  equi- 
distant from  these  sources,  the  two  systems  will  arrive  at  A 
in  the  same  phase  and  will  reenforce  each  other.  Hence 
on  the  screen  at  A  there  will  appear  a  bright  band  parallel 
to  the  refracting  edge  of  the  prism. 

Let  PI  be  so  situated  that  the  distance  PI  02  shall  ex- 
ceed Pi  0i  by  half  a  wave-length  for  the  homogeneous  light 
employed,  the  yellow  of  sodium,  for  example.  Then  the 
two  systems  will  meet  at  PI  in  exact  opposition  of  phase. 
This  is  the  condition  for  destructive  interference,  and  at 
PI  there  will  be  a  dark  band  parallel  to  the  bright  one 
through  A. 

At  P2  where  P202  exceeds  P20i  by  an  entire  wave- 
length there  will  be  a  bright  band  again  ;  and  still  further 
out  will  be  found  a  second  dark  band,  and  so  on.  We 

therefore   conclude  that  if   02P  —  0iP  —  n  - ,  where  X  is 

the  wave-length  of  the  light  employed,  there  will  be  a 
dark  band  for  all  odd  values  of  w,  and  a  bright  band  for 
even  values.  The  screen  will  then  be  illuminated  with 
a  series  of  bright  bands  alternating  with  dark  ones.  By 
measuring  the  distance  from  A  to  the  first  dark  band. 


INTERFERENCE    AND    DIFFRACTION.  299 

and  the  distance  A  0  it  is  possible  to  compute  the  differ- 
ence Pi  0  — PI#I,  and  so  to  measure  the  wave-length  of 
the  light.  A  measurement  of  this  kind  shows  that  the 
wave-length  of  yellow  sodium  light  is  only  5.89  x  10  ~~* 
cm.  Hence  the  vibration-frequency,  which  is  the  quotient 
of  the  speed  by  the  wave-length,  is  about  500  millions  of 
millions  a  second. 

Particular  attention  should  be  given  to  the  fact  that 
there  is  110  loss  of  energy  in  interference,  but  only  a  redis- 
tribution of  it  among  the  dark  and  light  bands. 

214.  Interference  produced  by  Thin  Films  (T.,  195  ;  D., 
5O3  ;  P.,  138;  B.,  492).  —  Interference  phenomena  are 
produced  by  thin  transparent  films.  The  iridescence  of 
ancient  glass,  of  a  thin  film  of  oil  on  water,  of  oxide  on  the 
surface  of  polished  or  molten  metal,  and  of  the  soap  bubble 
are  familiar  examples.  These  colors  are  the  residuum  of 
white  light  left  after  some  portions  have  been  cut  out  by 
interference  between  the  two  wave  systems  reflected  from 
the  parallel  surfaces  of  the  film. 
Let  A  A  (Fig.  148)  be  one  sur- 

face  of   the  film   and  BB   the      A _ 
other.     Light  incident   on   the      B  p.    m  B 

upper  surface  is  partly  trans- 
mitted and  partly  reflected.  The  portion  penetrating  the 
first  surface  is  in  part  reflected  from  the  second,  and 
emerges  again  from  the  first  surface  along  with  light  which 
has  undergone  reflection  only.  If  now  the  time  required 
to  traverse  the  film  twice  be  the  period  of  vibration  of  the 
homogeneous  light  employed,  then  the  system  which  has 
been  reflected  internally  will  on  emergence  be  in  the  same 
phase  as  the  system  reflected  externally,  so  far  as  difference 
of  path  is  concerned ;  and  if  difference  of  phase  depends 


300  LIGHT. 

only  on  difference  of  path  traversed,  when  this  difference 
vanishes  with  a  film  of  infinitesimal  thickness  the  two 
pencils  should  be  in  the  same  phase  and  the  illumination  a 
maximum.  But  the  fact  is  when  a  film  is  made  as  thin  as 
possible  it  becomes  black.  All  the  light  reflected  is  ex- 
tinguished by  interference.  The  difference  of  phase  there- 
fore depends  on  something  besides  difference  of  path.  This 
is  found  in  the  fact  that  the  two  reflections  take  place 
under  different  conditions,  one  in  the  rare  medium  next 
to  the  dense,  and  the  other  in  the  dense  medium  next  to 
the  rare.  One  of  the  two  interfering  systems  in  conse- 
quence loses  half  an  undulation  relative  to  the  other  in  the 
mere  act  of  reflection.  This  is  analogous  to  the  change  in 
phase  of  a  sound-wave  reflected  from  the  end  of  an  opiri 
organ  pipe.  When  a  condensation  is  reflected  from  the 
closed  end  of  a  pipe  there  is  a  change  of  sign  of  the  motion, 
but  not  of  the  condensation.  When  it  is  reflected  from  the 
end  of  an  open  pipe,**  there  is  no  change  of  sign  of  the 
motion,  but  the  condensation  changes  sign,  or  is  reflected  as 
a  rarefaction.  Therefore  if  two  condensations  could  be 
brought  together  after  reflection,  one  from  the  closed  end 
of  a  pipe  and  the  other  from  an  open  end,  the  two  dis- 
turbances would  be  found  in  opposite  phases  and  would 
interfere.  So  two  pencils  of  light  reflected  under  the 
corresponding  opposite  conditions  have  thereby  impressed 
upon  them  a  difference  of  phase  equal  to  half  a  period. 

When  therefore  the  thickness  of  the  film  and  the  angle 
of  incidence  are  such  that  one  pencil  falls  behind  the  other 
in  transmission  by  a  whole  number  of  wave-lengths,  inter- 
ference takes  place  with  extinction  of  light,  since  a  phase 
difference  of  half  a  period  must  be  added  because  of  the 
reflection  at  the  two  surfaces  under  opposite  conditions. 

With  white  light  the  extinction  of  one  spectral  color  by 


INTERFERENCE    AND    DIFFRACTION. 


301 


interference  leaves  colored  fringes.  It  is  to  be  observed 
that  the  thickness  of  film  that  would  produce  a  relative 
retardation  of  the  internally  reflected  pencil  of  one  wave- 
length for  violet  would  be  a  retardation  of  only  about  half 
a  wave-length  for  red  (217).  Therefore  extinction  of  both 
colors  cannot  take  place  at  the  same  part  of  the  film.  If 
the  violet  is  cut  out  the  red  remains.  Similar  reasoning 
applies  to  the  intermediate  colors.  The  reflected  light  is 
therefore  fringed  with  color. 

215.  Diffraction  Fringes  with  a  Narrow  Aperture 
(A.  and  B.,  443;  P.,  175;  B.,  501).  —  When  a  beam  of 
sunlight  is  admitted  through  a  very  narrow  slit  into  a 
darkened  room,  and  is  received  upon  a  screen  at  some 
distance,  there  will  be  seen  a  central  band  of  white  light 
in  the  direct  path  of  the  beam,  bordered  with  colored 
fringes.  Through  so  small  an  opening  light  passes  not 
merely  as  a  definite  pencil,  but  it  also  diverges  in  all 
directions  from  all  points  of  the 
opening  as  new  centres  of  disturb- 
ance. This  phenomenon  is  called 
diffraction.  The  waves  of  light 
thus  bend  around  an  obstruction, 
like  the  edge  of  the  slit,  in  the 
same  manner  as  water-waves  run 
around  an  obstruction.  The  colors 
will  be  intensified  if,  instead  of  a 
single  narrow  opening,  a  series  of 
equidistant  fine  lines  ruled  on 
smoked  glass  be  employed.  This  ng/i49. 

grating,  as  it  is  called,  should  be 

placed  at  the  focus  of  a  lens  through  which  a  beam  of 
sunlight  passes.  The  room 'must  be  well  darkened,  and 


302  LIGHT. 

all  extraneous  light  should  be  cut  off  by  properly  ar- 
ranged screens. 

Let  ab  (Fig.  149)  be  a  section  of  the  slit,  the  length  of 
which  is  perpendicular  to  the  plane  of  the  paper,  and  let 
MN\)Q  the  screen.  Then  r  in  the  direction  of  the  pencil 
is  nearly  equidistant  from  every  point  between  a  and  &, 
and  all  the  secondary  waves,  starting  from  the  several 
points  between  a  and-  b  as  centres,  arrive  at  r  in  nearly 
the  same  phase.  There  is,  therefore,  maximum  illumina- 
tion at  this  point.  Let  s  be  a  point  so  situated  that  the 
distance  as  shall  exceed  bs  by  one  wave-length  X  for 
some  color,  as  red.  Then  red  will  suffer  extinction  by  in- 
terference at  s.  For  if  ab  be  divided  into  two  equal  parts 
the  difference  in  distance  of  a  and  d  from  s  is  half  an 
undulation,  or  one-half  X;  the  secondary  waves  from  a 
and  d  meet  at  s  in  opposite  phases  ;  and  every  wave 
from  a  point  between  a  and  d  meets  at  s  a  wave  of  opposite 
phase  from  a  corresponding  point  between  d  and  b. 
Therefore  total  extinction  of  light  of  this  particular  wave- 
length X  takes  place  at  s. 

If  a  and  b  are  distant  from  a  point  on  the  screen  by 
g 

^X,  then  the  slit  may  be  divided  into  three  equal  parts; 
2 

the  secondary  waves  from  two  of  these  parts  interfere 
at  the  screen  as  explained,  while  those  from  the  third  part 
produce  illumination. 

In  general  if  as  —  bs  on  either  side  of  r  is  an  even 
number  of  half-waves  there  will  be  an  even  number  of 
half-period  elements  in  ab,  which  will  mutually  interfere 
at  s,  and  the  illumination  will  be  less  than  if  there  is  an 
odd  number  of  half-period  elements  in  ab.  With  mono- 
chromatic light,  bright  and  dark  bands  will  alternate  on 
either  side  of  r.  The  position  of  these  bands  is  given 


INTERFERENCE    AND    DIFFRACTION. 


303 


by  the  equation 


as  —  bs  =  n  — , 
2 


where  n  is  even  for  the  dark  and  odd  for  the  bright  bands. 
With  white  light,  in  which  X  is  different  for  the  differ- 
ent colors,  extinction  for  different  colors  will  take  place  at 
different  distances  of  s  from  r,' and  hence  colored  fringes 
will  appear  on  the  screen. 


216.  Spectrum  by  a  Diffraction  Grating  (A.  and  B., 
446;  P.,  186).  —  A  system  of  very  narrow,  equal,  and 
equidistant  rectangular  apertures  is  called  a  diffraction 
grating.  These  may  be  made  by  cutting  with  a  diamond 
point  by  means  of  a  dividing  engine  a  number  of  parallel 
equidistant  lines  on  a  glass  plate.  The  light  then  passes 
through  the  transparent  spaces  between  the  lines. 

Let  a  plane  wave  approach  the  grating  in  the  direction 
of  the  arrow  (Fig.  150).  Let  a,  c,  etc.,  be  the  parallel 
apertures,  and  let  parallel  lines 
ab,  cd,  etc.,  be  so  drawn  that  the 
distance  ae  to  the  foot  of  the  per- 
pendicular from  c  on  ab  shall  equal 
some  definite  wave-length  of  light 
X.  Then  an  will  be  an  exact 
number  of  wave-lengths  n\,  co  will 
be  (n  —  1)  X,  and  so  on.  The  line 
mn  will  therefore  touch  the  front 
of  a  series  of  secondary  waves  from  the  successive  trans- 
parent openings,  all  in  the  same  phase ;  and  if  this  new 
plane  wave  be  transmitted  through  a  lens  with  its  axis 
parallel  to  ab,  all  the  transparent  apertures  of  the  grating 
will  send  light  to  the  focus  in  the  same  phase  and  of  wave- 
length X. 


Fig.  150. 


304  LIGHT. 

Let  6  be  the  angle  between  ab  and  the  direction  of  the 
light  incident  on  AB.  Then 

ae  =  X  =  d  sin  #, 

where  d  is  the  distance  ae  from  centre  to  centre  of  the 
openings.  The  disturbance  at  the  focus  of  the  lens  will 
be  the  resultant  of  all  the  disturbances  coming  from  the 
numerous  apertures  of  the  grating,  and  all  those  of  wave- 
length d  sin  0  will  arrive  in  the  same  phase. 

For  disturbances  of  any  other  wave-length  this  coinci- 
dence in  phase  will  not  exist.  If,  for  example,  the  differ- 
ence between  X  and  d  sin  9  is  rio^  then  the  light  from 
the  first  opening  will  be  opposite  in  phase  to  that  from  the 
fifty-first,  that  from  the  second  will  be  in  opposition  to 
that  from  the  fifty-second,  etc.  Hence  all  the  light  from 
the  first  fifty  openings  will  exactly  neutralize  that  from  the 
second  fifty  in  the  direction  6.  Since  the  number  of  lines 
on  the  grating  is  several  thousand  to  a  centimetre,  light 
of  only  one  wave-length  is  found  at  any  angle  6  with  the 
direction  of  the  incident  beam.  Hence  a  pure  or  normal 
spectrum  is  produced  in  which  with  a  given  grating  the 
angular  separation  of  the  different  colors  depends  only 
on  their  wave-length. 

From  the  formula  X  =  d  sin  6  it  is  obvious  that  the 
longest  waves  are  found  at  the  greatest  deviations  ;  for 
the  first  spectrum  the  wave-lengths  are  nearly  proportional 
to  the  deviations,  for  01  is  small.  Hence  there  can  be  no 
irrationality  of  dispersion  in  a  diffraction  spectrum. 

For  greater  deviations  we  may  have 

2X  =  d  sin  <92 . 

Such  waves  produce  a  spectrum  of  the  second  order 
lying  farther  away  from  the  normal  to  the  grating.  For 
the  spectrum  of  the  third  order 


INTERFERENCE    AND    DIFFRACTION.  305 

3\  =  d  sin  03 , 
and  for  the  nth  spectrum 

n\  =  d  sin  6n. 

These  spectra  are  all  violet  at  the  inner  edge  nearest 
the  normal  to  the  grating  and  red  at  the  outer.  Over- 
lapping of  spectra  will  occur  when  the  deviation  for  the 
violet  of  any  order  is  less  than  that  of  the  preceding  red. 
Moreover,  since  the  deviation  is  approximately  propor- 
tional to  wave-length,  and  the  wave-length  of  red  is  some- 
what less  than  twice  that  of  violet,  the  red  of  the  second 
spectrum  will  overlap  the  violet  of  the  third.  The  separa- 
tion of  the  superposed  parts  may  be  effected  by  means  of 
a  prism. 

The  diffraction  grating  furnishes  an  admirable  method 
of  measuring  wave-lengths.  The  distance  d,  whicli  is  a 
constant  of  the  grating,  must  be  measured,  and  the  spec- 
trometer must  be  supplied  with  a  graduated  circle  for  the 
purpose  of  measuring  the  deflection  0,  corresponding  to 
any  dark  line  of  the  solar  spectrum,  or  to  any  bright  line 
of  an  artificial  spectrum  of  a  vapor. 

Spectra  similar  to  the  preceding  may  be  obtained  by 
reflection  from  a  grating  ruled  with  very  fine  parallel 
grooves  on  speculum  metal.  The  surface  must  first  be 
finely  polished.  The  pencils  reflected  from  the  polished 
intervals  between  the  rulings  come  as  if  from  virtual 
images  of  the  source  through  the  intervals  of  the 
grating.  The  exquisite  colors  of  mother-of-pearl  and 
other  striated  surfaces,  of  the  feathers  of  certain  birds, 
and  of  changeable  silk  are  instances  of  the  same  method 
of  producing  colors  from  white  light  by  diffraction. 

Professor  Rowland's  famous  reflecting  gratings  are  ruled 
on  concave  surfaces  of  polished  speculum  metal. 


806 


LIGHT. 


217.  Wave-lengths  and  Vibration-frequencies.  —  The 
unit  employed  in  measuring  wave-lengths  of  light  is  the 
tenth-metre,  of  which  1010  are  required  to  make  a  metre. 
The  following  are  the  values  for  the  principal  Fraunhofer 
lines  in  air  at  20°  C.  and  760  mm.  pressure : 


B  . 

6884.11 

C  . 

.  .  6563.07 

5896.18 

j) 

5890.22 

E! 5270.52 

E2 5269.84 

F 4861.51 

Q 4293. 

Hi      .     .     .  3968. 


Taking  the  speed  of  light  as  300  million  metres  a  second, 

or  300  x  1016  tenth-metres,  the  vibration-frequencies  corre- 

sponding to  the   above  spectral  lines  may  be  found  by 

dividing   this   speed   by  the    several    wave-lengths,    since 

V 


The  result  is  as  follows  : 


A    .     .     . 

C   . 


393.6  X 

435.8 

457.1 

508.8- 
509.3 


K)1 


569.2  X  1012 


F 617.1 

O 698.8 

H, 756.0 


Thus  the  light  entering  the  eye  and  producing  the  violet 
color  represented  by  S^  is  due  to  756  millions  of  millions 
of  vibrations  a  second.  A  photograph  of  the  sun  has  been 
taken  with  an  exposure  of  only  one  twenty-thousandth 
of  a  second.  But  in  this  time  a  beam  of  light  15,000 
metres  (9.32  miles)  in  length  has  entered  the  camera, 
and  fully  375  x  108  or  37,500  millions  of  waves  have  im- 
pressed their  effects  on  the  sensitized  plate. 


INTERFERENCE    AND    DIFFRACTION.  307 

The  visible  spectrum  lies  between  wave-lengths  of  about 
7500  and  3900  tenth-metres.  Rowland  has  measured  them 
from  7714.657  to  3094.736  tenth-metres,  and  has  photo- 
graphed them  from  about  7000  to  3000.  Langley  has 
measured  lunar  radiations  with  wave-lengths  of  170,000 
tenth-metres,  or  nearly  twenty-three  times  as  long  as  the 
longest  waves  exciting  vision.  The  invisible  spectrum 
extends  several  times  the  length  of  the  visible  spectrum 
beyond  the  extreme  violet;  so  that  the  entire  invisible 
spectrum  actually  explored  is  perhaps  thirty  times  as  long 
as  the  visible  one.  Physically  the  only  difference  existing 
among  these  radiations  is  one  of  wave-length.  All  of  them 
represent  energy  which  is  converted  into  heat  when 
absorbed  by  the  proper  surfaces ;  and  perhaps  all  may  be 
able  to  excite  or  precipitate  chemical  changes  if  the  proper 
sensitive  substances  are  found  for  different  parts  of  the 
spectrum.  The  mechanism  of  the  eye  limits  its  receptivity 
to  the  visible  spectrum.  The  differences  formerly  sup- 
posed to  exist  between  the  so-called  light,  heat,  and  actinic 
rays  are  therefore  differences  in  the  receptive  apparatus. 


308  LIGHT. 


CHAPTER    XII. 
COLOR 

213.  Modes  of  producing  Color.  —  Color  has  no  ob- 
jective existence,  but  is  the  physiological  character  as- 
signed to  light  by  sensation.  The  only  physical  differences 
corresponding  to  different  colors  are  differences  in  wave- 
length. Color  in  light  corresponds  to  pitch  in  sound,  with 
this  difference,  that  an  indefinite  number  of  color  mixtures 
may  produce  the  same  effect  on  the  eye,  while  the  ear 
analyzes  a  complex  tone  into  its  elements  and  recognizes 
the  intervals.  Hence  each  combination  of  sounds  pro- 
duces its  own  effect. 

Red  is  due  to  the  longest  ether-waves  exciting  vision, 
and  violet  to  the  shortest.  The  production  of  colors  from 
white  light  involves,  therefore,  some  process  of  isolating 
vibrations  of  certain  definite  periods.  These  processes  are 
three  in  number : 

(a)     Refraction. 

(5)     Interference. 

(c)     Absorption. 

The  analysis  of  white  light  into  its  component  colors  by 
means  of  the  first  two  methods  has  already  been  described. 
It  remains  to  explain  briefly  the  third. 

219.  Color  of  Opaque  Bodies  (L.,  180;  T.,  147;  P., 
384). — All  bodies,  except  those  with  highly  polished 
surfaces,  reflect  light  by  irregular  reflection  from  greater 


COLOR.  309 

or  less  depths  within  them.  If  all  the  components  of 
white  light  are  reflected  in  the  same  proportion  the  body 
appears  white  or  gray.  Such  is  the  case  with  a  sheet  of 
white  paper  or  a  white  screen  on  which  the  solar  spectrum 
is  projected.  These  surfaces  reflect  diffused  light  in  all 
directions  and  without  preference  for  light  of  particular 
wave-lengths.  Hence  all  the  colors  of  the  spectrum  on 
such  a  screen  appear  the  same  as'when  they  are  received 
directly  into  the  eye  placed  in  the  path  of  the  diverging 
beam  from  a  prism.  But  if  the  body  exhibits  any  inequal- 
ity in  its  relative  absorbing  and  reflecting  power  for  light 
of  different  wave-lengths  lying  within  the  visible  spectrum, 
then  it  will  appear  colored  when  white  light  is  incident 
upon  it,  the  color  being  the  result  of  mixing  those  color 
components  of  white  light  which  the  body  reflects.  The 
other  spectral  colors  are  absorbed.  A  mixture  of  these 
would  produce  a  color  complementary  to  that  of  the  reflected 
components.  Complementary  colors  are  those  which,  added 
together,  produce  the  impression  of  white  light. 

The  colors  of  natural  objects  are  therefore  chiefly  resid- 
uals left  after  absorption. 

It  is  easy  to  justify  this  conclusion  by  appropriate  ex- 
periments. Let  a  solar  spectrum,  with  rather  wide  slit,  be 
projected  on  a  white  screen  by  means  of  a  carbon  disulphide 
prism.  Take  a  flower  which  shows  rich  red  petals  in  large 
masses,  such  as  a  tulip, -or  certain  geraniums.  Hold  it  in 
the  spectrum  and  pass  it  through  the  different  colors.  In 
the  red  the  flower  shines  with  its  usual  bright  red  color ; 
but  as  it  is  moved  along  into  the  green  it  becomes  black 
and  continues  to  show  no  power  of  reflection  for  the  re- 
mainder of  the  spectrum.  All  the  other  colors  except  red 
are  readily  absorbed.  The  red,  on  the  contrary,  is  reflected 
and  gives  the  color  to  the  body.  We  see  therefore  that 


310  LIGHT. 

the  body  can  exhibit  no  color  not  already  present  in  the 
light  which  illuminates  it.  A  piece  of  red  flannel  is 
brilliantly  red  in  the  less  refrangible  end  of  the  spectrum, 
but  suddenly  turns  to  a  dirty  brown,  and  then  a  dead  black, 
when  moved  out  of  the  red  toward  the  violet.  Ribbons  of 
various  colors  carried  along  through  the  spectrum  give 
very  instructive  results,  which  are  readily  explained  by  the 
power  of  selective  absorption  possessed  by  them. 

The  essential  nature  of  the  colors  of  objects  may  be 
indicated  by  saying  that  they  are  the  residue  of  the  light, 
by  which  they  are  illuminated,  after  abstraction  of  the  rays 
extinguished  by  absorption. 

With  homogeneous  illumination  differences  of  color  are 
no  longer  possible.  This  fact  is  strikingly  exhibited  by 
viewing  objects  of  various  colors  in  a  room  lighted  only 
with  burning  sodium.  The  most  healthful  face  presents 
an  ashen  hue,  and  brilliant  flowers  are  reduced  to  a  faded 
yellow.  "  Were  the  sun  a  sphere  of  glowing  vapor  of 
sodium,  all  terrestrial  nature  would  present  this  monoto- 
nous and  gloomy  aspect.  It  requires  the  white  light  of  the 
sun,  in  which  innumerable  colors  are  blended,  to  disclose 
to  our  eyes  the  variegated  tints  of  natural  objects." 

220.  Color  of  Transparent  Bodies  (T.,  151;  P.,  379; 
L.,  172).  — If  a  transparent  body  absorbs  all  radiations 
within  the  visible  spectrum  in  equal  proportion,  it  is  color- 
less ;  but  if  it  is  transparent  to  certain  radiations  affecting 
the  eye  and  not  to  others,  it  appears  colored  by  transmitted 
light,  and  the  color  is  due  to  the  mixed  impression  pro- 
duced by  the  transmitted  radiations. 

The  different  colors  of  transparent  bodies  result  from 
their  individual  capabilities  of  selective  absorption.  This 
generalization  is  easily  justified  by  means  of  the  spectrum. 


COLOR.  311 

Plants  owe  their  color  to  the  chlorophyll  contained  in 
their  cells.  An  alkaline  solution  of  this  coloring  matter, 
placed  in  the  prismatic  beam,  produces  a  deep  black  band 
in  the  middle  of  the  red,  only  feeble  absorption  striae  in 
the  yellow  and  green,  while  the  entire  indigo-violet  part 
of  the  spectrum  from  about  the  middle  onward  is  entirely 
absent.  The  light  transmitted  by  chlorophyll  is  there- 
fore red  and  predominantly  green. 

If  a  piece  of  blue  cobalt  glass  be  interposed  in  the  solar 
beam,  the  spectrum  will  consist  of  a  small  amount  of  the 
extreme  red  and  all  of  the  indigo-violet  part  which  the 
chlorophyll  absorbs. 

A  piece  of  glass,  colored  red  with  the  sub-oxide  of  copper, 
allows  only  the  red  and  orange-red  rays  as  far  as  the 
D  line  to  pass  through.  All  the  rest  of  the  spectrum  is 
completely  stopped  by  this  glass.  If  now  the  light  be 
passed  through  the  cobalt-blue  and  the  copper-red  glasses 
in  succession,  the  only  part  of  the  spectrum  surviving  the 
double  absorption  process  is  the  extreme  dark  red  trans- 
mitted by  both. 

A  solution  of  potassium  bichromate  transmits  the  less 
refrangible  part  of  the  spectrum  only.  Its  spectrum  stops 
at  the  Fraunhofer  line  b.  A  solution  of  the  ammoniated 
oxide  of  copper  transmits  only  the  more  refrangible  part 
of  the  spectrum  from  the  b  line  on.  These  two  colors, 
therefore,  contain  all  the  spectral  tints,  and  are  comple- 
mentary to  each  other.  But  if  the  two  solutions  in  flat 
glass  cells  be  placed  in  the  path  of  a  beam  of  sunlight,  one 
behind  the  other,  the  combination  scarcely  permits  the 
passage  of  any  light.  The  one  fluid  looked  at  through 
the  other  appears  almost  perfectly  black.  The  light  that 
struggles  through  the  one  is  stopped  by  the  other.  It 
must  not  be  supposed  that  this  experiment  illustrates  a 


312  LIGHT. 

mixture  of  colors  or  colored  lights.  Far  from  it !  It  illus- 
trates successive  absorption  by  transparent  bodies. 

If  the  blue  ammoniated  copper  oxide  solution  be  placed 
in  front  of  the  yellow  solution  of  normal  potassium  chromate 
of  the  proper  density,  the  light  transmitted  by  the  two  will 
be  green.  If  their  separate  spectra  be  examined,  it  will 
be  found  that  both  solutions  transmit  green.  Hence  green 
is  the  only  color  common  to  the  two,  and  is  therefore  the 
only  one  which  is  not  stopped  by  absorption  in  the  one 
solution  or  the  other. 

The  same  explanation  applies  to  the  green  color  obtained 
by  mixing  yellow  and  blue  pigments.  The  light  penetrates 
below  the  surface  of  thin  layers  of  such  pigments  and  suf- 
fers absorption  during  transit  through  them.  Hence  green 
is  the  only  color  which  survives  the  double  process  of 
absorption.  A  mixture  of  pigments  is  not  a  mixture 
of  colored  lights.  It  is  rather  a  process  of  successive 
absorption;  the  resulting  color  is  the  residue. 

221.  Mixing  Colored  Lights.  —  In  order  to  perceive 
the  mixed  effect  due  to  two  or  more  colors,  it  is  neces- 
sary that  they  fall  upon  the  retina  either  simultaneously 
or  in  quick  succession.  Visual  impressions  persist  for  a 
small  fraction  of  a  second ;  and,  if  one  remains  till  the 
arrival  of  another,  both  impressions  are  simultaneously 
present. 

If  two  partially  overlapping  discs  of  light  be  projected 
on  the  screen  and  transparent  colored  bodies  be  placed  in 
the  path  of  the  two  beams,  the  light  reflected  to  the  eye 
from  the  overlapping  area  will  consist  of  a  real  mixture  of 
the  two  colored  beams.  Thus  if  the  ammoniated  copper 
oxide  solution  be  placed  in  the  path  of  one  beam  and  the 
potassium  chromate  in  the  other,  the  area  common  to  the 


COLOR.  313 

two  discs  will  be  white  or  gray  with  the  proper  density 
of  the  two  solutions.  When  their  colored  images  are 
added  they  cannot  in  any  way  be  made  to  produce  green. 

So  the  cobalt  blue  and  the  oxide  of  copper  red  glasses 
will  give  beams  of  light  which  by  addition  produce  white. 

If  a  disc  of  cardboard,  colored  in  sections,  be  rapidly 
rotated,  the  result  is  a  mixture  or  superposition  of  visual 
impressions.  But  very  different  mixtures  may  produce 
the  same  visual  impression.  The  eye  has  no  power  of 
analyzing  light  into  its  constituents.  "Any  one  of  the 
elementary  colors,  from  the  extreme  red  to  a  certain  point 
in  the  yellowish  green,  can  be  combined  with  another 
elementary  color  on  the  other  side  of  the  green  in  such 
proportion  as  to  yield  a  perfect  imitation  of  ordinary 
white."  The  unaided  eye  can  tell  nothing  about  the  com- 
position of  colored  light ;  it  must  be  studied  by  means  of 
the  spectroscope,  armed  with  a  prism  or  a  diffraction 
grating. 

222.  Three  Primary  Color  Sensations  (B.,  479  ;  D., 
529).  —  The  Youiig-Helmholtz  theory  of  color  sensations 
supposes  that  each  element  of  the  retina,  broad  enough  to 
perceive  light,  consists  of  three  ultimate  nerve-ends,  each 
of  which  serves  to  give  perception  of  one  of  three  physio- 
logically primary  colors.  All  other  color  perceptions  are 
due  to.  the  simultaneous  excitation  of  these  three  sets  of 
nerve-ends  in  varying  relative  degrees.  The  three  primary 
color  sensations  are  red,  green,  and  violet,  though  appar- 
ently any  three  such  colors  may  be  made  the  basis  of  a 
systematic  classification  of  colors. 

The  theory  supposes  that  one  set  of  nerve  terminals  gives 
red  when  excited  by  waves  of  long  period ;  the  second  gives 
green,  when  excited  by  waves  of  intermediate  period ;  and 


314  LIGHT. 

the  third  gives  violet  when  excited  by  waves  of  short 
period.  When  the  first  and  second  set  are  both  stimulated 
the  resulting  sensation  is  yellow ;  when  the  second  and 
third  are  alone  stimulated  the  result  is  a  sensation  of  blue. 
An  equal  stimulation  of  all  three  produces  the  impression 
of  white  light. 

Hence  red  and  green  light  mixed  produces  on  the  eye 
the  impression  of  yellow,  while  the"  spectroscope  shows  at 
the  same  time  the  entire  absence  of  the  spectral  color 
yellow.  Yellow  as  a  sensation  may  therefore  be  produced 
either  by  one  stimulus  of  a  definite  period,  which  excites 
both  the  first  and  second  sets  of  nerve-ends ;  or  by  two 
stimuli,  corresponding  to  red  and  green,  exciting  the  two 
sets  of  nerve-ends  simultaneously.  .  So  green  and  violet 
give  blue.  In  general  colors  near  each  other  in  the  spec- 
trum give  when  compounded  an  intermediate  color  sensa- 
tion. 

Since  yellow  contains  both  the  red  and  green  sensations 
or  stimuli,  and  blue  contains  both  the  green  and  violet, 
when  yellow  and  blue  are  compounded  they  contain  all 
three  color  sensations,  and  produce  the  impression  of  white. 

The  visual  apparatus  of  the  eye  is  not  affected  by  radia- 
tions below  the  extreme  red,  nor  by  those  above  the  ex- 
treme violet.  The  blindness  of  the  eye  to  very  long  and 
very  short  ether-waves  may  be  an  advantage.  The  energy 
of  the  former  is  so  great  that  ordinary  vision  would  be 
impossible  if  our  eyes  responded  to  their  stimulus ;  while 
if  the  ultra  violet  rays  produced  vision  chromatic  aberra- 
tion would  be  excessive,  and  clear  images  would  be  impos- 
sible. What  spectral  colors  would  be  added  to  the 
chromatic  scale  by  an  extension  of  vision  to  either  the 
infra  red  or  the  ultra  violet  waves  we  have  no  moans  of 
conjecturing. 


COLOR.  315 

223.  Subjective  Colors.  —  The  theory  of  primary  color 
sensations  furnishes  a  ready  explanation  of  colors  due  to 
fatigue  of  the  retina.  It  is  well  known  that  objects  are 
quite  invisible  to  one  entering  a  faintly  lighted  room  after 
exposure  of  the  eyes  for  a  little  time  to  a  bright  light.  After 
being  subjected  to  a  strong  stimulus  the  eye  loses  its  sen- 
sitiveness to  a  weak  one.  This  liability  to  fatigue  is  char- 
acteristic not  only  of  the  retina  as  a  whole,  but  of  any  portion 
of  it  giving  one  of  the  primary  color  sensations.  Fatigue  of 
the  retina  causes  it  to  lose  the  power  of  responding  to  the 
stimulus  of  any  color  long  looked  at ;  and  when  the  eye  is 
then  directed  toward  a  moderately  illuminated  white  sur- 
face it  appears  of  a  tint  complementary  to  the  one  which  has 
produced  the  fatigue.  This  is  one  form  of  subjective  colors. 
If  the  eye  be  fixed  for  half  a  minute  on  a  colored  picture, 
red,  for  example,  in  a  strong  light,  and  be  then  directed 
to  a  less  strongly  illuminated  white  wall,  an  image  of  the 
picture  will  be  seen  in  green,  enlarged  if  the  wall  is  more 
distant  than  the  picture  itself.  The  eye,  fatigued  for  red, 
still  maintains  its  sensitiveness  for  green.  Hence  the 
relatively  faint  white  light  is  sufficient  to  stimulate  for 
green  while  it  furnishes  insufficient  stimulus  for  red.  The 
state  of  the  retina  therefore  serves  to  abstract  from  white 
light  a  sensation  due  to  a  portion  of  its  component  radi- 
ation, leaving  the  rest.  This  is  a  fourth  method  of  color 
production,  but,  unlike  the  other  three,  it  is  not  objective. 
It  belongs  more  especially  to  physiology  than  to  physics. 

It  has  been  shown  that  when  the  eye  is  fatigued  with 
white  light  it  recovers  its  sensibility  for  different  colors  suc- 
cessively after  different  intervals  of  time.  If  one  looks  out 
of  a  small  window,  such  as  a  porthole,  on  a  strongly  illumin- 
ated sky,  and  then  closes  one's  eyes,  the  after  image  of  the 
window  will  appear  in  dissolving  colors  of  brilliant  hues. 


316  LIGHT. 

Simultaneous  color  contrasts  are  another  form  of  sub- 
jective colors.  These  are  well  displayed  by  laying  thin 
tissue  paper  over  black  letters  printed  on  a  ground  colored 
green.  The  letters  in  a  strong  light  are  pink  by  contrast. 
The  tissue  paper  furnishes  a  faint  illumination  in  contrast 
with  the  strong  green,  and  the  unfatigued  nerve  terminals 
giving  red  cause  the  letters  to  appear  red  or  pink  in  con- 
trast with  the  complementary  green.  Hence  certain  colors 
are  heightened  by  contrast,  particularly  complementary 
ones. 


POLARIZED    LIGHT.  317 


CHAPTER    XIII. 

POLARIZED    LIGHT. 

224.  Polarization  (P.,  232;  D.,  476;  S.,  1;  L,,  293). 
—  In  the  study  of  light  up  to  this  point  there  has  been  no 
occasion  to  inquire  respecting  the  direction  of  vibration  in 
a  wave  of  light.  In  sound  the  particles  of  the  atmosphere 
have  a  longitudinal  motion,  and  many  of  the  phenomena 
of  wave-motion  in  sound  are  helpful  in  the  study  of 
analogous  phenomena  in  the  theory  of  light,  irrespective 
of  any  differences  growing  out  of  longitudinal  as  opposed 
to  transverse  vibrations. 

But  we  now  approach  a  class  of  optical  phenomena  of 
the  highest  interest,  to  which  the  theory  of  sound  furnishes 
no  parallels.  Such  phenomena  belong  to  what  is  known 
as  polarized  light. 

If  a  plate  of  red,  brown,  or  green  tourmaline,  cut  parallel 
to  its  optic  axis,  is  held  so  that  a  beam  of  light  falls  upon 
it  normally,  the  transmitted  beam  will  be  found  to  'have 
undergone  a  most  remarkable  change.  To  the  unaided 
eye  the  transmitted  light  differs  in  no  respect  from  the 
incident  beam,  except  that  it  has  undergone  a  slight 
change  in  color  on  account  of  the  natural  tint  of  the 
particular  tourmaline  crystal  used. 

But  if  the  transmitted  beam  be  examined  by  means  of  a 
piece  of  plate  glass,  it  will  be  found  that  while  in  one 
direction  it  is  reflected  in  the  same  manner  as  common 
light,  yet  when  the  glass  is  turned  around  the  beam  as  an 


318  LIGHT. 

axis  the  light  is  found  to  vary  in  intensity  ;  and  in  one  posi- 
tion the  reflected  light  vanishes  entirely.  The  light  trans- 
mitted by  the  tourmaline  can  be  reflected  in  one  plane  at 
all  angles,  but  in  the  plane  at  right  angles  to  this  it  is  im- 
perfectly reflected,  and  at  a  certain  angle  of  incidence  is 
not  reflected  at  all. 

Further,  if  the  beam  transmitted  through  one  crystal  of 
tourmaline  be  examined  by  a  second  parallel  plate,  cut  in 
the  same  way,  it  will  be  found  that  in  one  position  of  the 
second  plate  the  light  is  freely  transmitted,  while  it  be- 
comes feebler  and  feebler  if  either  plate  be  turned  around 
in  its  own  plane ;  and  when  the  two  parallel  plates  have 
their  longer  dimensions  at  right  angles,  no  light  what- 
ever passes  through  them.  The  light  is  completely 
extinguished  by  crossing  two  transparent  and  nearly 
colorless  crystals.  The  beam  of  light  transmitted  by 
the  first  plate  possesses  a  kind  of  two-sidedness,  analogous 
to  the  two-endedness  of  a  magnet. ;  hence  the  analogous, 
though  unfortunate,  name  of  polarization,  which  is  applied 
to  it. 

Such  a  beam  of  light  is  said  to  be  plane  polarized.  The 
plate  of  tourmaline  is  called  the  polarizer ;  and  the  plate 
glass  reflector,  or  the  second  tourmaline,  the  analyzer. 
The  one  brings  the  light  into  the  condition  of  polarization 
and  the  other  serves  to  examine  it.  There  are  many 
processes,  some  natural  and  others  artificial,  by  means  of 
which  light  may  be  polarized ;  and  the  apparatus  serving 
as  a  polarizer  may  always  be  used  as  an  analyzer. 

It  is  important  to  observe  that  the  proportion  of  the 
beam  transmitted  by  the  polarizer,  which  also  passes 
through  the  analyzer,  depends  upon  the  orientation  of  the 
second  plate  with  respect  to  the  first,  and  when  the  two 
plates  are  crossed  the  beam  transmitted  by  the  first  is 


POLARIZED    LIGHT.  319 

completely  extinguished  by  the  second.  Now,  if  the  vibra- 
tions were  longitudinal,  it  is  impossible  to  conceive  how 
the  crossing  of  the  plates  could  stop  the  light,  since  the 
rotation  of  the  second  plate  could  not  affect  them.  The 
phenomenon  of  extinction  is  therefore  held  to  demonstrate 
that  the  vibrations  of  the  ether  constituting  light  (and 
radiant  energy  in  general)  are  transverse. 

In  plane  polarized  light  the  vibrations  are  confined  to  a 
single  plane.  Light  may  also  be  circularly  polarized.  The 
motions  of  the  ether  particles  are  then  confined  to  circles 
about  the  ray  as  an  axis,  successive  particles  having  equal 
difference  of  phase. 

If  the  ether  particles  rotate  in  ellipses  about  the  ray  as 
an  axis,  the  light  is  said  to  be  elliptically  polarized. 

The  relation  of  circular  and  elliptical  polarization  to 
two  systems  of  plane  vibrations,  at  right  angles  to  each 
other,  will  be  readily  understood  when  the  composition 
of  simple  harmonic  motions  at  right  angles  and  of  the 
same  period  is  recalled  (Art.  130). 

225,  Double  Refraction  in  Iceland  Spar  (D.,  5O9  ;  B., 
510;  P.,  236;  S.,  17;  L.,  282;  T.,  205).  —  In  the  study 
of  refraction  up  to  this  point  it  has  been  assumed  that 
light  travels  with  one  velocity  in  the  denser  transparent 
medium,  and  that  there  is  consequently  but  one  refracted 
beam  for  each  incident  one.  But  all  crystalline  substances, 
except  those  whose  fundamental  form  is  the  cube,  possess 
the  property  of  producing  two  refracted  beams.  This 
property  is  called  double  refraction.  It  belongs  also  to 
animal  and  vegetable  substances  having  a  regular  arrange- 
ment of  parts,  and  to  transparent  media,  like  glass, 
unequally  strained  in  different  directions. 

If  a  crystal  of  Iceland  spar  (crystallized  carbonate  of 


320  LIGHT. 

calcium)  be  laid  on  a  printed  page,  the  letters  will  in  gen- 
eral appear  double ;  or  if  a  beam  of  sunlight  be  admitted 
through  a  small  round  opening  in  the  shutter  and  focused 
on  a  screen,  and  if  a  crystal  of  Iceland  spar  be  placed  in  the 
path  of  the  beam  so  that  the  beam  is  incident  normally  on 
its  surface,  two  equally  illuminated  discs  will  appear  on  the 
screen.  The  incident  beam  is  divided  into  two  by  the  pro- 
cess of  double  refraction. 

One  of  these  beams  as  it  emerges  from  the  spar  follows 
precisely  the  same  course  as  if  it  had  traversed  a  piece  of 
glass,  and  is  therefore  called  the  ordinary  ray. 

The  other  beam,  on  the  contrary,  is  laterally  displaced  in 
a  direction  depending  on  the  position  of  the  crystal ;  and 
if  the  crystal  be  rotated,  keeping  the  angle  of  incidence 
normal,  the  spot  belonging  to  the  ordinary  ray  will  remain 
fixed,  while  the  other  one  will  describe  a  circle  around  it. 
Further,  if  the  angle  of  incidence  be  changed,  the  refrac- 
tion of  the  second  beam  does  not  follow  the  law  of  sines. 
Hence  this  one  is  called  the  extraordinary  ray. 

In  every  doubly  refracting  crystal  there  is  at  least  one 
direction  in  which  no  bifurcation  of  the  ray  takes  place. 
This  direction  is  called  an  optic  axis  of  the  crystal.  The 
refracted  rays  diverge  most  widely  when  the  incident  beam 
is  perpendicular  to  the  optic  axis.  In  Iceland  spar  the 
ordinary  <av,  which  follows  the  law  of  single  refraction, 
has  an  index  of  refraction  for  the  D  line  of  1.6585.  The 
extraordinary  ray  does  not  in  general  lie  in  the  plane  of 
incidence,  and  its  refractive  index  varies  from  1.6585  to 
1.4865.  The  minimum  value  is  called  the  extraordinary 
index. 

If  a  crystal  of  the  spar  be  examined  it  will  be  found  that 
the  solid  angles  at  two  opposite  corners  are  contained 
between  three  obtuse  plane  angles.  The  direction  making 


POLARIZED  LIGHT.  821 

equal  angles  with  the  planes  of  these  obtuse  angles  is  called 
the  optic  axis  of  the  crystal.  Iceland  spar  has  but  one 
such  axis,  and  is  therefore  called  uniaxial.  Any  plane 
normal  to  a  refracting  face  of  the  crystal  and  parallel  to 
the  optic  axis  is  called  a  principal  plane.  If  the  two  solid 
angles  bounded  by  the  three  obtuse  angles  are  cut  away 
by  planes  perpendicular  to  the  optic  axis,  a  beam  incident 
normally  on  either  of  these  surfaces  passes  through  in  the 
direction  of  the  optic  axis  without  bifurcation. 

If  the  plane  of  incidence  is  perpendicular  to  the  optic 
axis,  both  rays  follow  the  ordinary  law  of  refraction,  and 
the  index  of  refraction  for  the  extraordinary  ray  is  then 
1.4865  for  sodium  light. 

A  prism  may  be  cut  from  Iceland  spar  with  its  refract- 
ing edge  parallel  to  the  optic  axis.  A  ray  traversing  it 
in  the  direction  of  minimum  deviation  is  perpendicular  to 
the  optic  axis,  and  the  two  components  have  the  maximum 
separation. 

Crystals  like  Iceland  spar,  in  which  the  extraordinary 
index  is  smaller  than  the  ordinary,  are  called  negative  uni- 
axial  crystals.  Quartz  also  produces  double  refraction ; 
but  in  quartz  the  angular  separation  of  the  two  rays  is 
smaller  than  in  the  spar,  and  the  indices  of  refraction  are 
for  sodium  light  1.544  for  the  ordinary  ray,  and  1.553  for 
the  extraordinary.  Such  crystals,  in  which  the  index 
of  the  ordinary  ray  is  the  smaller  of  the  two,  are  called 
positive  uniaxial  crystals. 

226.  Theory  of  Double  Refraction  (T.,  2O7 ;  P.,  247; 
B.,  513  ;  A.  and  B.,  475  ;  L.,  286  ;  D.,  511).  —  Since  the 
refractive  index  is  inversely  as  the  speed  of  light  in  any 
medium,  it  follows  that  for  doubly  refracting  substances 
there  are  two  speeds  of  light.  For  the  ordinary  ray  the 


322 


LIGHT. 


speed  is  the  same  in  every  direction,  while  for  the  extraor- 
dinary ray  the  speed  varies  with  the  direction  of  the 
ray  between  limits  proportional  to  1.486  and  1.658.  It 
has  been  found  for  Iceland  spar  that  the  phenomena  can 
be  completely  represented  by  supposing  that  a  disturbance, 
started  in  the  interior  of  a  crystal,  gives  rise  to  two  con- 
centric wave  surfaces,  one  spherical  and  the  other  a  flat- 
tened ellipsoid,  with  its  polar  diameter  parallel  to  the  optic 
axis,  and  equal  to  the  diameter  of  the  sphere.  The  polar 
and  equatorial  diameters  are  as  1.486  to  1.658,  or  as  0.603 
to  0.676.  The  two  wave  surfaces  therefore  touch  at  the 
extremities  of  the  polar  diameter,  and  the  spherical  surface 
of  the  ordinary  ray  lies  wholly  within  the  ellipsoid  of 
revolution.  A  crystal  in  which  the  wave  surfaces  are  thus 
related  is  a  negative  uniaxial  crystal.  If  the  sphere  en- 
closes the  ellipsoid,  the  two  touching  at  the  poles  of  the 
ellipsoid,  the  crystal  is  a  positive  uniaxial  crystal. 

From  these  two  wave  surfaces  the  path  of  the  two  rays 
may  be  determined  by  the  method   already  employed  in 

Art.  187.  Thus  let 
ic  (Fig.  151)  be  the 
direction  of  the  in- 
cident light  falling 
on  a  crystal  of  Ice- 
land spar,  AB.  Let 
the  plane  of  inci- 
dence contain  the 
R  optic  axis  ab.  With 
c  as  a  centre,  and 
with  ic  produced  as  the  major  axis,  draw  a  circle  and  an 
ellipse  with  major  and  minor  axes  proportional  to  the  ordi- 
nary and  extraordinary  indices  of  refraction,  the  radius  of 
the  circle  being  the  semi-minor  axis.  Draw  also  a  larger 


Fig.  151. 


POLARIZED   LIGHT.  323 

circle  with  a  radius  equal  to  the  distance  light  will  travel 
in  air  while  the  ordinary  ray  travels  over  the  radius  of  the 
small  circle  in  the  spar.  Then  if  nm  is  tangent  to  the  outer 
circle  at  the  point  of  intersection  with  ic  produced,  it  is 
parallel  to  the  incident  wave;  draw  tangents  from  m  to 
the  inner  circle  and  to  the  ellipse ;  the  lines  co,  ce,  con- 
necting c  with  the  points  of  tangency,  give  the  direction 
of  the  ordinary  and  extraordinary  rays  respectively. 

It  is  evident  that  if  the  optic  axis  were  perpendicular 
to  the  plane  of  the  paper  through  <?,  the  section  of  the  ellip- 
soid made  by  the  plane  of  incidence  would  also  be  a  circle^ 
the  extraordinary  ray  would  lie  in  the  plane  of  incidence 
for  all  angles  of  incidence,  and  would  have  its  least  re- 
fractive index  and  its  greatest  speed  of  transmission. 

227.  Polarization  by  Double  Refraction  (A.  and  B., 
476;  D.,  513;  B.,  517). --The  examination  of  light 
transmitted  by  Iceland  spar,  either  by  another  crystal  or 
by  a  piece  of  unsilvered  plate  glass,  exhibits  a  marked 
difference  between  it  and  ordinary  light.  Let  the  ex- 
traordinary ray  be  intercepted  by  a  screen,  and  let  the 
ordinary  ray  fall  on  the  plate  glass  at  an  angle  of  inci- 
dence of  57°.  If  now  the  plane  of  incidence  coincides 
with  the  principal  plane  of  the  spar,  the  light  will  be  re- 
flected like  ordinary  light ;  but  if  the  mirror  is  rotated 
about  the  beam  of  light  as  an  axis  the  reflected  light  will 
grow  dimmer  and  dimmer ;  and  when  the  plane  of  inci- 
dence is  at  right  angles  to  the  principal  plane  of  the 
spar,  the  light  will  fail  altogether.  If  the  rotation  is  con- 
tinued, the  light  gradually  regains  its  maximum  intensity 
at  180°,  and  again  fails  at  270°.  The  extraordinary  ray 
exhibits  the  same  peculiarities  in  the  same  order,  but  it  lias 
its  maximum  brightness  at  90°  and  270°,  and  fails  at  0° 


824  LIGHT. 

and  180°.  Both  rays  are  therefore  plane  polarized,  the 
ordinary  in  the  plane  of  the  principal  section,  and  the  ex- 
traordinary in  a  plane  at  right  angles  thereto.  The  vibra- 
tions composing  the  ordinary  ray  are  considered  to  be  at 
right  angles  to  the  optic  axis,  while  those  of  the  extraor- 
dinary ray  are  in  a  plane  containing  the  optic  axis  and  the 
incident  ray,  and  may  make  any  angle  with  the  optic 
axis  from  0°  to  90°. 

The  plane  of  incidence  in  which  the  light  is  most  freely 
reflected  is  called  the  plane  of  polarization.  The  vibra- 
tions composing  the  reflected  ray  are  parallel  to  the  re- 
flecting surface.  -^ 

228.  Polarization  by  Reflection  (T.,  213;  A.  and  B., 
48O;  P.,  234).  --When  light  has  been  reflected  from 
such  surfaces  as  water,  glass,  polished  wood,  etc.,  at  a 
definite  angle  depending  upon  the  nature  of  each  sub- 
stance, it  is  found  to  possess  all  the  properties  of  light 
polarized  by  Iceland  spar  or  tourmaline.  For  glass  there 
is  a  particular  angle  of  incidence  at  which  the  reflected 
light  is  completely  polarized,  and  this  is  called  the  angle, 
of  polarization.  Brewster  discovered  that  for  this  angle 
the  reflected  and  refracted  rays  are  at  right  angles  to  each 
other.  The  tangent  of  the  angle  of  incidence  then  equals 
the  index  of  refraction.  For  if  the  reflected  and  refracted 
rays  are  at  right  angles  the  corresponding  angles  are  com- 
plementary, or 

sin  r  —  cos  i. 

m       £  sin  i      sin  i 

Therefore          p  =  --  .  =  tan  i. 

sin  r      cos  i 

It  must  not  be  inferred,  however,  that  for  every  sub- 
stance there  is  an  angle  of  complete  polarization.  The 
polarization  always  increases  with  the  angle  of  incidence 


POLARIZED  LIGHT.  325 

up  to  a  maximum  and  then  decreases  again,  after  passing 
the  angle  of  maximum  polarization.  This  maximum  is 
called  the  polarizing  angle  of  the  substance.  Only  a  few 
substances,  with  a  refractive  index  of  about  1.46,  polarize 
light  completely  by  reflection.  If  the  substance  is  trans- 
parent the  refracted  ray  is  also  polarized,  and  in  a  plane 
perpendicular  to  that  of  the  reflected  ray.  The  plane  of 
polarization  for  the  latter  is  the  plane  of  incidence,  and  its 
vibrations  are  parallel  to  the  reflecting  surface. 

229.  Nicol's  Prism  (A.  and  B.,  481 ;  P.,  254;  S.,  22). 
—  A  single  beam  of  plane  polarized  light  may  be  pro- 
duced by  transmission  through  a  bundle  of  parallel  plates 
at  an  angle  of  incidence  of  about  57°,  the  polarizing  angle 
for  glass  ;  also  by  the  passage  through  a  plate  of  tourmaline 
cut  parallel  to  its  optic  axis.  Tourmaline  is  a  doubly  re- 


4  Fig.    152. 

fracting  substance  and  has  the  property  of  rapidly  absorb- 
ing the  ordinary  ray,  so  that  a  plate  1  or  2  mm.  thick  is 
impervious  to  it.  The  extraordinary  ray,  on  the  contrary, 
it  transmits.  Tourmaline  is  not,  however,  a  very  trans- 
parent, material,  and  the  most  effective  arrangement  for 
securing  a  beam  of  plane  polarized  light  is  a  Nicol's  prism. 
It  is  constructed  of  Iceland  spar  in  such  a  way  that  one  of 
the  refracted  rays  is  stopped  by  total  internal  reflection. 
A  long  rhomb  of  the  spar  has  its  terminal  faces  cut  off 
obliquely  so  that  the  angle  ACE  (Fig.  152)  is  68°.  The 


326  LIGHT. 

plane  of  the  figure  is  a  principal  plane.  The  rhomb  is  then 
cut  through  by  a  plane,  the  trace  of  which  is  AB,  making 
the  angle  ABC  22°.  The  two  faces  of  the  section  are 
polished  and  cemented  together  with  Canada  balsam,  which 
has  an  index  of  refraction  intermediate  between  those  of 
the  ordinary  and  the  extraordinary  rays. 

When  therefore  a  ray  of  light  ab  enters  the  prism  it  is 
divided  into  two  rays,  bo  the  ordinary,  and  be  the  extraor- 
dinary. But  bo  meets  the  Canada  balsam  at  an  angle 
somewhat  greater  than  68°,  while  the  critical  angle  for  the 
ordinary  ray  is  67°  31'.  The  relative  index  of  refraction 
for  the  ordinary  ray  from  Canada  balsam  to  Iceland  spar  is 

1    £\E\  Q 

—  '-—  -  (Art.  187),  1.532  being  the  index  of  the  balsam  from 

air.    But  the  sine  of  the  critical  angle  is  the  reciprocal  of  the 

1  532 

relative  index,  or  .  ~V     —  0.924.    This  is  the  sine  of  67°  31'. 

l. 


Therefore  the  ordinary  ray  suffers  total  internal  reflec- 
tion in  a  Nicol's  prism.  The  critical  angle  for  the  extraor- 
dinary ray  from  Canada  balsam  to  Iceland  spar,  for  this 
angle  of  incidence,  is  greater  than  68°.  It  is  not  reflected 
at  the  first  surface  of  the  balsam,  because  it  goes  from  a 
medium  of  lower  refractive  index  for  it  to  one  of  higher  ; 
and  it  is  not  reflected  at  the  second  surface  of  the  balsam 
because  the  angle  of  incidence  is  less  than  the  critical 
angle  for  the  two  media.  Moreover,  since  the  cemented 
section  is  at  right  angles  to  a  principal  plane  of  the  crys- 
tal, the  vibrations  parallel  to  this  section,  and  therefore 
those  readily  reflected,  constitute  the  ordinary  ray.  Thus 
the  extraordinary  ray  alone  passes  through. 

The  direction  of  vibration  for  the  transmitted  ray  is  the 
shorter  diagonal  of  the  end  of  the  prism.  This  is  in  a 
principal  plane.  A  Nicol's  prism  thus  permits  only  those 


POLARIZED    LIGHT.  327 

vibrations  to  traverse  it  which  are  in  its  principal  plane, 
while  it  is  completely  opaque  to  vibrations  at  right  angles 
to  its  principal  plane. 

230.  Extinction   of  Light   by   two   Crossed  Nicol's 
Prisms. — When  the  light  which  has  passed  through  one 
Nicol's  prism  falls  upon  a  second,  the  amount  transmitted 
will  depend  upon  the  relation  of  the  principal  planes  of 
the  two.     The  first  prism  is  called  the  polarizer,  and  the 
second  the  analyzer.    If  their  shorter  diagonals  are  parallel, 
then  the  plane  polarized  light  from  the  polarizer  will  com- 
pose the  extraordinary  ray  in   the  analyzer,  and  will  pass 
on  through   unaffected.     But   if   the  analyzer  be  turned 
around  the  beam  of  light  as  an  axis,  the  transmitted  beam 
will  decrease  in  brightness,   and  will    disappear  entirely 
when  the  rotation  has  reached   90°.     The  Nicol's  prisms 
are  then  said  to  be  "  crossed  "  ;  the  light  from  the  polarizer 
now  forms  the  ordinary  ray  for  the  analyzer,  and  is  lost  by 
internal   reflection.     In   intermediate   positions   the  recti- 
linear vibrations  of  the  plane  polarized  extraordinary  ray 
are  resolved  into  two  rectangular  components  in  directions 
corresponding  with  the   two  planes  of    vibration  in   the 
analyzer.     This  resolution  takes  place  in  accordance  with 
the  usual  mechanical  law  for  the  resolution  of  a  motion 
into  two  rectangular  components. 

The  intensity'of  the  transmitted  light  is  proportional  to 
the  cos2a,  a  being  the  angle  through  which  the  analyzer 
has  been  rotated  from  the  position  of  parallelism  with  the 
polarizer. 

231.  Effect  of  Interposing  a  Doubly  Refracting  Plate 
(T.,  227;  A.  and  B.,  483;  B.,  528;  L.,  316;  S.,  28).  - 

If  two  Nicol's  prisms  be  placed  with  their  principal  planes 


328 


LIGHT. 


crossed  no  light  will  pass  through  them.  Suppose  now  a 
thin  doubly  refracting  plate  of  mica  or  selenite  to  be  in- 
serted between  them.  It  will  be  found  that  there  are  two 
positions  of  the  plate  at  right  angles  to  each  other  in  which 
the  field  will  remain  dark.  In  all  other  positions  of  the 
mica  plate  the  light  will  be  restored,  reaching  its  maximum 
intensity  when  the  plate  has  turned  round  in  its  own  plane 
45°  from  the  positions  of  no  effect. 

The  mica  or  selenite  is  a  doubly  refracting  substance  ; 
and  when  the  plate  is  in  a  position  such  that  its  two  direc- 
tions of  vibration  coincide  with  the  principal  planes  of 
the  polarizer  and  analyzer,  the  extraordinary  ray  from  the 
polarizer  passes  through  without  resolution  into  two  compo- 
nents ami  is  stopped  by  the  analyzer.  But  in  any  other 
position  of  the  plate  the  case  is  different.  Within  it  the 
rectilinear  vibrations  of  the  plane  polarized  ray  will  be 
divided  into  two  components.  Let  Ox  and  Oy  (Fig.  153) 
be  the  two  directions  of  vibration  for  the 
mica  or  selenite  plate.  Then  if  OA  rep- 
resent the  semiamplitude  of  vibration 
for  the  incident  ray,  it  will  be  resolved 
into  two  vibrations  whose  semiamplitudes 
a;  are  OM  and  ON.  Both  of  these  rays 
will  pass  through  the  plate  unchanged 
except  that  one  travels  faster  than  the 
other ;  a  difference  of  'phase  will  there- 
fore result,  dependent  upon  the  thick- 
ness of  the  plate.  The  recombination  of 
-  these  two  vibrations  on  emergence  will 
result  in  elliptic  vibration  if  the  tAvo  beams  produced  by 
the  mica  are  not  separated  so  far  that  they  do  not  overlap. 
But  wherever  the  same  ether  elements  are  disturbed  by 
the  vibrations  of  both  the  rays  they  will  describe  elliptical 


M 


N 


O 


Fig    153. 


POLARIZED   LIGHT.  329 

orbits,  and  all  the  possible  ellipses  will  be  inscribed  within 
the  rectangle  AA'BB1  (Art.  131).  The  amplitudes  of  the 
motion  parallel  to  Ox  and  Of/  are  not  altered,  but  the 
maximum  displacement  in  one  direction  is  no  longer 
simultaneous  with  that  in  the  other,  unless  the  differ- 
ence of  phase  becomes  some  multiple  of  27r.  The  light 
emerging  from  the  thin  plate  is  then  in  general  elliptically 
polarized.  If  OM  equals  ON  and  the  difference  of  phase 
becomes  an  odd  multiple  of  £TT,  the  light  will  be  circularly 
polarized. 

OA  must  then  be  inclined  at  an  angle  of  45°  with  Ox, 
and  the  thickness  of  the  plate  must  be  such  that  one  of 
the  component  vibrations  suffers  a  relative  retardation  of 
one-quarter  of  a  wave-length  in  traversing  it ;  for  the  dis- 
tinctive feature  of  double  refraction  is  the  difference  of 
speed  of  the  two  rays. 

Suppose  now  that  the  elliptically  polarized  light  passes 
on  to  the  analyzer.  The  introduction  of  the  mica  plate  in 
general  restores  the  light.  If  the  mica  plate  is  made  to 
rotate  in  its  own  plane  the  light  vanishes  for  successive 
positions  differing  by  a  quadrant  of  rotation.  In  these 
positions  the  directions  of  vibration  for  the  interposed  crys- 
tal coincide  with  the  principal  planes  of  the  Nicol's  prisms ; 
and  the  light  from  the  first  prism  passes  unchanged 
through  the  crystal  and  is  extinguished  by  the  second  prism. 
Midway  between  these  positions  of  extinction  the  light 
transmitted  by  the  system  is  brightest.  If  the  mica  is  of 
such  thickness  as  to  produce  circular  polarization,  the 
rotation  of  the  analyzer  does  not  alter  the  brightness  of 
the  transmitted  light.  When  the  elliptically  polarised 
ray  enters  the  analyzer,  each  of  the  two  rectilinear  compo- 
nents of  the  elliptical  motion  is  resolved  in  the  two  direc- 
tions of  vibration  for  the  analyzer.  One  pair  of  these 


330  LIGHT. 

components  unite  to  form  the  ordinary  ray,  which  is  ex- 
tinguished; the  other  pair  form  the  extraordinary  ray, 
which  is  transmitted. 

232.  Colors  produced  by  Polarized  Light  (P.,  244; 
S.,  37;  L.,  319).  —  Colors  produced  from  white  light  by 
means  of  polarization  are  due  to  destructive  interference. 
But  interference   cannot  take  place  between  rays  whose 
vibrations  are  at  right  angles  to  each  other.     The  office  of 
the  analyzer  is  to  bring  together  into  one  plane  one  com- 
ponent from  each  pair  into  which  it  resolves  the  two  rays 
from  the   doubly   refracting    plate.       Two    rays    of    light 
polarized  at  right  angles  do  not  interfere  like  two  rays  of 
ordinary  light. 

Of  the  two  pairs  of  components  of  the  elliptically  polar- 
ized light  entering  the  second  Nicol's  prism,  the  one  forming 
the  extraordinary  ray  exhibits  interference  if  the  two  com- 
ponents have  opposite  phases.  The  other  pair  then  have  the 
same  phase ;  and  if  the  analyzer  is  rotated  through  90°, 
they  form  the  extraordinary  ray  and  are  transmitted.  If 
now  the  light  is  not  homogeneous,  then,  as  the  difference 
in  phase  depends  upon  wave-length,  the  retardation  of  one 
component  compared  with  the  other  is  such  as  to  produce 
complete  interference  for  a  definite  wave-length  and  partial 
interference  for  waves  of  approximately  the  same  length. 
The  corresponding  colors  then  suffer  complete  or  partial 
extinction,  while  the  remaining  colors  of  the  incident  light 
are  transmitted,  forming  a  colored  beam. 

233.  Complementary   Colors   in   the   Two   Standard 
Positions  of  the  Analyzer  (A.  and  B.,  485;  S.,  35;  L., 
32O).  — Who  n  the  principal  plane  of  the  lamina  of  selenite 
(crystallized  sulphate  of  calcium)  forms   an  angle  of  45° 


POLARIZED   LIGHT.  331 

with  the  plane  of  vibration  of  both  polarizer  and  analyzer, 
the  most  brilliant  colors  are  obtained  with  a  thin  film.  If 
now  the  analyzer  be  turned  through  90°  into  parallelism 
with  the  polarizer,  complementary  colors  of  nearly  equal 
brilliancy  will  appear. 

If  the  components  along  Ox  (Fig.  153)  annul  each  other, 
the  color  to  which  they  correspond  is  wanting  in  the  plane 
polarized  light  vibrating  in  this  direction  ;  but  at  the  same 
time  the  components  along  Oy  are  added,  and  the  same 
color  is  found  undiminished  in  the  light  whose  vibrations 
are  confined  to  this  plane.  For  other  colors  "  the  relative 
retardation  is  different ;  but  for  each  vibration  period,  the 
component  in  the  direction  Ox  combined  with  that  in  the 
direction  Oy  represents  the  total  light  for  that  period  in 
the  beam  entering  the  analyzer."  Hence  the  sum  of  the 
two  represents  all  the  light  entering  the  analyzer;  and 
therefore  the  light  transmitted  when  the  Nicol's  prisms  are 
crossed  must  be  complementary  to  that  passing  when  they 
are  parallel,  if  the  incident  light  is  white. 

Thick  plates  of  sejenite  do  not  produce  color.  If  the 
doubly  refracting  plate  is  thick  enough  to  produce  a  rela- 
tive retardation  of  several  wave-lengths  for  extreme  violet, 
it  will  produce  a  retardation  of  half  as  many  wave-lengths 
for  red,  and  an  intermediate  number  for  intermediate  colors. 
Hence  with  crossed  prisms  extinction  of  these  colors  will 
take  place.  These  losses  will  be  distributed  at  about  equal 
distances  along  the  spectrum.  But  the  transmitted  light 
will  consist  of  the  different  colors  in  nearly  the  same  pro- 
portion as  in  white  light,  and  it  will  therefore  be  white, 
but  of  diminished  intensity. 

If  two  plates  of  selenite,  of  exactly  the  same  thickness, 
and  therefore  producing  the  same  tint,  are  superposed  in 
such  a  manner  that  their  principal  planes  coincide,  or  so 


332  LIGHT, 

that  the  extraordinary  ray  through  the  one  is  also  the 
extraordinary  ray  through  the  other,  they  exhibit  another 
color,  which  is  precisely  the  same  as  that  produced  by  a 
plate  of  double  the  thickness  of  either.  But  if  they  are 
superposed  so  that  their  principal  planes  are  perpendicular 
to  each  other  they  produce  no  effect.  The  screen  remains 
dark.  The  ray  which  travels  the  more  slowly  in  the  first 
lamina  travels  the  more  rapidly  in  the  second,  and  the  two- 
emerge  together  as  if  a  plate  of  annealed  glass  had  been 
interposed.  The  two  rays  which  leave  the  compound 
plate  have  no  difference  of  phase,  and  cannot  exhibit 
interference  and  color.  If  the  crossed  laminae  are  of 
unequal  thickness,  the  effect  is  the  same  as  that  produced 
by  a  single  lamina  whose  thickness  is  their  difference. 

234.  Colored  Rings  produced  by  a  Plate  cut  at  Right 
Angles  to  the  Optic  Axis  (L.,  326;  S.,  95).  —  Extremely 
beautiful  and  interesting  phenomena  are  produced  by 
plates  of  uniaxial  crystals  cut  perpendicular  to  the  optic 
axis,  such  as  a  section  of  Iceland  spar,  in  a  beam  of 
converging  plane  polarized  light.  The  central  ray  of  the 
converging  cone  should  be  normal  to  the  plate.  It  then 
passes  through  the  crystal  along  the  optic  axis  without 
undergoing  double  refraction.  But  all  other  rays  in  the 
cone  traverse  the  crystal  section  obliquely  and  are  doubly 
refracted.  The  further  the  ray  is  from  the  axis  of  the 
cone  the  greater  is  the  obliquity  of  its  path  through  the 
plate  and  the  greater  the  thickness  traversed.  The  re- 
tardation of  the  one  component  behind  the  other  is  also 
greater ;  and  since  at  equal  distances  from  the  optic  axis 
both  causes  determining  difference  of  path  of  the  two 
doubly  refracted  rays  are  equal,  it  follows  that  the  same 
difference  of  path  must  exist  for  all  points  of  a  circle  con- 


POLARIZED   LIGHT.  333 

ceived  as  drawn  upon  the  screen  around  the  intersection 
with  it  of  the  axial  ray  of  the  cone.  Hence  a  system  of 
concentric  colors  appears  on"  the  screen  in  iridescent  rings 
like  those  of  Newton's  rings,  obtained  by  pressing  the 
convex  side  of  a  plano-convex  lens  against  a  plate  of  plane 
glass. 

When  the  polarizer  and  analyzer  are  crossed  the  colored 
rings  are  traversed  by  a  black  cross.  This  is  explained  as 
follows  :  Since  the  optic  axis  is  perpendicular  to  the  surface 
of  the  plate  of  spar,  every  straight  line  drawn  through  the 
centre  of  the  system  of  rings  is  the  trace  of  a  principal 
plane.  The  vibrations  of  the  ordinary  ray  are  perpendicu- 
lar to  a  principal  plane  and  therefore  tangential  to  all  the 
concentric  circles  ;  those  of  the  extraordinary  ray  are  in  a 
principal  plane  or  radially  in  the  circles.  Hence  in  the 
two  diameters,  representing  the  planes  of  vibration  of 
the  analyzer  and  polarizer,  the  directions  of  vibration  in 
the  thin  plate  correspond  with  those  of  the  polarizer  and 
analyzer,  and  therefore  in  these  two  directions  the  thin 
plate  produces  no  double  refraction  and  has  no  effect  on 
the  light.  In  all  other  directions,  the  tangential  and  radial 
directions  of  vibration  for  the  plate  are  inclined  to  those 
of  the  polarizer  and  analyzer,  and  therefore  double  refrac- 
tion takes  place  with  interference  and  colors. 

If  the  analyzer  is  turned  so  as  to  be  parallel  with  the 
polarizer  a  white  cross  takes  the  place  of  the  black  one. 
The  reason  is  evident,  the  colored  rings  being  then  simply 
projected  on  a  bright  field  as  a  background. 

235.  Double  Refraction  in  Quartz  (A.  and  B.,  491 ;  S., 
41).  —  Quartz  is  a  positive  uniaxial  crystal  and  gives  an 
ordinary  and  an  extraordinary  ray.  When  a  quartz  plate, 
cut  perpendicular  to  the  optic  axis,  is  interposed  between 


334  LIGHT. 

the  polarizer  and  analyzer,  the  effects  differ  greatly  from 
those  produced  by  Iceland  spar  or  by  selenite.  Let  the 
plane  polarized  light  fall  on  the  quartz  normally.  Then 
with  crossed  Nicol's  prisms,  the  light  is  restored  and  is 
unchanged  by  the  rotation  of  the  quartz  through  any 
azimuth.  Homogeneous  light  will  be  extinguished  by  the 
rotation  of  the  analyzer  through  a  certain  angle,  indicating 
that  the  effect  of  the  quartz  is  to  rotate  the  plane  of  polar- 
ization. The  amount  of  the  rotation  is  the  same  as  the 
angle  through  which  the  analyzer  must  be  turned  to  pro- 
duce extinction.  Some  specimens  of  quartz  rotate  the 
plane  of  polarization  to  the  right,  in  relation  to  the  direc- 
tion of  the  light,  and  are  therefore  called  right-handed. 
Others  rotate  it  to  the  left,  and  are  called  left-handed.  The 
amount  of  rotation  depends  upon  the  thickness  of  the 
plate  and  the  wave-length  of  the  light.  Hence  with  white 
light  the  effect  of  rotating  the  analyzer  is  to  quench  in 
succession  the  several  colors  as  the  plane  of  polarization 
for  each  is  reached.  The  resulting  colored  beam  changes 
its  tint  continuously  as  the  analyzer  rotates.  For  a 
given  plate  the  angle  of  rotation  of  the  plane  of  vibration 
varies  nearly  inversely  as  the  square  of  the  wave-length. 
For  a  quartz  plate  one  mm.  thick  Brock  found  that  the 
B  line  was  rotated  through  15°  18',  and  the  G  line,  42°  12'. 
In  the  explanation  of  these  phenomena  given  by  Fres- 
nel,  the  vibrations  of  the  two  rays  for  quartz  are  supposed 
not  to  be  rectilinear  but  circular,  and  in  opposite  directions. 
The  light  of  each  ray  is  circularly  polarized.  The  circular 
motion  of  the  ether  is  right-handed  for  one  ray,  and  left- 
handed  for  the  other.  When  these  two  motions  are  im- 
pressed upon  the  same  portions  of  the  ether  at  the  same 
tfme,  the  result  is  plane  polarized  light  (Art.  32).  But 
one  of  these  motions  is  transmitted  with  greater  speed 


POLARIZED   LIGHT.  335 

than  the  other ;  in  other  words,  its  period  of  motion  in  the 
circle  is  slightly  less ;  and  since  the  resulting  simple  har- 
monic motion  is  in  the  plane  of  symmetry,  that  plane 
rotates  in  the  direction  of  the  component  motion  of  shorter 
period.  The  rotation  of  the  plane  of  polarization  of  the 
resulting  plane  polarized  light  is  in  the  same  direction  as 
that  of  the  plane  of  symmetry. 

•  Many  liquids,  including  a  solution  of  sugar,  rotate  the 
plane  of  polarization.  An  instrument  for  comparing  this 
rotation  with  that  produced  by  quartz  is  called  a  saccha- 
rimeter. 


INDEX. 


Numbers  refer  to  pages. 


Aberration,  chromatic,  288  ;  spher- 
ical, 254,  282 ;  of  light,  234. 

Absorption,  308;  double  process 
of,  312;  selective,  310. 

Academy,  French,  4. 

Acceleration,  12;  angular,  85,  86; 
centripetal,  30;  in  simple  har- 
monic motion,  33  ;  linear,  85  ;  of 
gravity,  16;  proportional  to  dis- 
placement, 33. 

Accelerations,  composition  of,  20. 

Achromatic  prisms,  289 ;  system, 
291. 

Achromatism,  291 ;  conditions  of, 
289. 

Activity,  52. 

Adiabatic  expansion,  151. 

Air  has  weight,  126. 

Air-pump,  130. 

Amplitude,  32,  154,  156,  160. 

Analyzer,  318,  327. 

Angle,  critical,  263,  326 ;  of  con- 
tact, 121,  124;  of  incidence, 
242;  of  reflection,  242;  of 
polarization,  324 ;  refracting, 
267. 

Angular,  acceleration,  85,  86;  ve- 
locity, 31,  84. 

Antinodes,  185,  206. 

Aplanatic  surface,  283. 

Archimedes,  principle  of,  109,  111. 

Astro-Physics,  2. 


Atmosphere,  height  of  the  homo=> 

geneous,  128. 

Atmospheric  pressure,  126,  132. 
Attraction,    'molecular,     116;     of 

gravitation,  116. 
Auxiliary  circle,  32. 
Axis,  for  minimum  period,  98;  of 

shadow,  232  ;  of  suspension,  96  ; 

optic,  320,  321,  332;  principal, 

250,  252,  256,  280. 

Bach,  181. 

Back  pressure  on  discharge  vessel, 

104. 

Balance,  sensibility  of,  75. 
Barometer,  128. 
Baume's  hydrometer,  113. 
Beats,  dissonance  due  to,  220  ;  in- 

terference and,  159;  phenome- 

non of,  161. 
Black  cross,  333. 
Boundaries  of  Physics,  1. 
Boyle's  law,  128,  148,  150. 
Bradley's  method,  234. 
Brock,  334. 
Buoyancy,   centre   of,    110;    cor- 

rection for,  131. 
Bureau  des  Lonitudes, 


Canada  balsam,  326. 
Capillary   action,    114; 
and  depression,  121. 


144. 


elevation 


338 


INDEX. 


Caustic  surface,  253,  2G1. 

Caustics  by  reflection,  253 ;  by  re- 
.  fraction,  261,  283. 

Centimetre,  4. 

Centre  of  buoyancy,  110;  of  in- 
ertia, 79 ;  of  oscillation,  9G ;  of 
percussion,  96;  of  pressure, 
106;  optical,  278. 

Chemistry,  2. 

Chladni,  194,  197. 

Chlorophyl,  311. 

Chord,  major  and  minor,  177. 

Chromatic  aberration,  288,  314. 

Circle,  of  least  confusion,  283;  of 
reference,  32. 

Circular  motion,  uniform,  29. 

Circularly  polarized  light,  319, 
329. 

Coefficient,  of  elasticity,  147,  148  ; 
o,f  expansion,  150. 

Colladon  and  Sturm,  152. 

Colloids,  115. 

Color,  308 ;' complementary,  309, 
316 ;  contrasts,  316 ;  by  polarized 
light,  330;  of  opaque  bodies, 
308;  of  the  spectrum,  205;  of 
transparent  bodies,  310;  sub- 
jective, 315;  sensations,  313. 

Colored  rings,  332. 

Comma,  179. 

Complementary  colors,  309,  316. 

Components,  18. 

Composition,' of  motions,  18;  of 
simple  harmonic  motions,  157, 
162,  163,  165,  166;  of  simulta- 
neous circular  motions,  38. 

Compression,  142;  adiabatic,  151; 
isothermal,  148. 

Concave,  grating,  305 ;  spherical 
mirrors,  249,  255. 

Condensation,  142,  150,  202. 


I  Conjugate  focal  distances,  273; 
foci,  251. 

Conservation,  of  energy,  59 ;  of 
matter,  60. 

Consonance,  221. 

Construction,  at  a  single  surface, 
265 ;  for  deviation,  269  ;  for  im- 
ages in  a  converging  lens,  280 ;  in 
a  diverging  lens,  282  ;  of  a  spher- 
ical wave  at  a  plane  surface, 
266;  for  refracted  ray,  262. 

Contact,  angle  of,  121,  124. 

Converging  lens,  275;  construc- 
tion for  image  in,  280. 

Convex  lens,  277;  spherical  mir- 
rors, 249. 

Cornu,  237. 

Corrections  for  temperature,  149. 

Couples,  71;  moment  of,  71. 

Critical  angle,  263,  267,  326. 

Cross,  black,  333. 

Crystalloids,  115. 

Crystals,  negative,  321 ;  positive, 
321 ;  uniaxial,  322. 

Current,  electric,  140. 

Curvature,  9,  266. 

Curve  of  sines,  154,  156. 

Dark  lines  in  solar  spectrum,  293. 
Density,  42;  and  specific  gravity, 

110;  of  liquids,  111 ;  of  mercury, 

110;  of  water,  110. 
Derived  units.  4. 
|    Deviation,  by  rotation  of  mirror, 

249;   construction   for,   2('>9 ;    in 

prisms,  268  ;  minimum,  268,  271 ; 

without  dispersion,  289. 
Diapason,  195. 
Diatonic  scale,  177. 
Direct  and  reflected  systems,  172. 
Direct-vision  spectroscope,  289. 


INDEX. 


839 


Differential  tones,  218. 
Diffraction,    297 ;     fringes,     301  ; 

grating,  303. 
Dispersion,  285;  irrationality  of, 

292;     of    energy,    64;    without 

deviation,  289. 

Dispersive  power,  287,  289,  291. 
Dissonance  clue  to  beats,  220. 
Distortion  of  image,  282. 
Dolland,  288. 
Double  refraction  in  Iceland  spar, 

319;  in  quartz,  333;  polarization 

by,  323. 
Double  weighing,  77;  refraction, 

321. 

Doubly  refracting  plate,  327. 
Dynamics,  6. 
Dyne,  44. 

Echo,  173;  aerial,  174. 

Eclipse,  of  Jupiter's  satellites, 
234 ;  in  Eizeau's  experiment,  237. 

Efflux,  velocity  of,  133. 

Elasticity,  coefficient  of,  147 ; 
equals  pressure,  148. 

Elevation  and  range,  25. 

Elliptically  polarized  light,  319, 
329. 

Elongation,  33. 

Energy,  54 ;  and  surface  tension, 
120 ;  availability  of,  63 ;  conser- 
vation of,  59 ;  independent  of 
elasticity,  141;  kinetic,  56;  not 
a  force,  57 ;  potential,  56 ;  trans- 
formations of,  61. 

Entropy,  64. 

Epoch,  33. 

Ether,  226;  disturbance  in,  227; 
existence  of,  226 ;  physics  of,  6; 
universal  medium,  226;  extraor- 
dinary ray,  320,  326. 


Film,  curved,  122  ;  liquid,  120,  125. 

Fizeau's  method,  235. 

Fluid,  mechanics  of,  99 ;  perfect, 
99;  pressure,  103,  105. 

Focal  length,  273. 

Foci,  conjugate,  251. 

Focus,  principal,  251,  273. 

Force,  43;  law  of  assumed,  115; 
practical  unit  of,  44. 

Forces,  parallel,  69. 

Formation  of  images  in  concave 
mirror,  252. 

Foucault's  experiment,  260 ;  meth- 
od, 238. 

Fourier,  law  of,  191 ;  theorem,  215. 

Fraunhofer,  294,  306. 

Free,  fall  of  bodies,  16;  surface 
of  liquids,  103. 

Fresnel,  298,  334. 

Fulcrum,  71.  * 

Function,  of  the  time,  23,  41 ; 
periodic,  41,  155.  .. 

Galileo,  126;  inclined  plane,  27. 
Geometrical  optics,  228. 
Gramme,  defined,  5. 
Graphical  representation  of  work, 

52. 
Grating,  concave,  305 ;  diffraction, 

303 ;  Rowland's,  305. 
Gravitation,  attraction  of,  116. 
Gravity,  acceleration  of,    16,    17, 

93;   centre  of,  79;  effect  of,  118. 
Gyration,  radius  of,  86. 

Harmonics,  191. 
Head,  pressure,  135. 
Hearing,  sound  and,  137. 
Heat-energy,  63. 

Helmholtz,  201,  212,  214,  217,  218, 
220,  313. 


340 


INDEX: 


Henry,  Joseph,  115. 

Hertz,  220. 

Homogeneous   atmosphere,    128; 

light,  298  ;  media,  228. 
Hooke's  law,  154. 
Horse-power,  52. 

Huyghens,  principle  of,  169. 
Hydraulic  press,  101. 
Hydrometers,  Baume's,  113;  gen- 
eral theory  of,  112. 

Iceland  spar,  319,  322. 

Images,  and  object  at  fixed  dis- 
tance, 277;  distortion  of,  282; 
formation  of,  252;  in  a  plane 
mirror,  243;  in  lenses,  275;  of 
images,  246;  perverted,  230; 
produced  by  small  apertures, 
229 ;  virtual,  244,  256. 

Impulse,  44,  59. 

Inclined  plane,  27,  74. 

Index,  absolute,  258 ;  extraordi- 
nary, 320;  of  refraction,  258; 
ordinary,  320 ;  relative,  258,  326. 

Inertia,  centre  of,  79,  81,  83;  il- 
lustrated, 47 ;  law  of,  47 ;  mo- 
ment of,  84,  86,  87,  88,  89,  90. 

Interference  and  beats,  159;  and 
diffraction,  297;  by  thin  films, 
299  ;  destructive,  330 ;  of  sound- 
waves, 160. 

Internal  reflection,  203,  ;-*25. 

Interval,  musical,  170. 

Inversion,  230. 

Irrationality  of  dispersion,  292. 

Isochronous  vibrations,  94. 

Isotropic  media,  228. 

Jew's-harp,  195. 
Jupiter's  satellites,  232. 


Kater's  reversible  pendulum,  97. 
Kinematics,  6,  8. 

Kinetic   energy,   56 ;  in  terms  of 
mass  and  velocity,  58  ;   of  rota- 
•  tion,  85. 

Kinetics,  7;  definition  of,  42. 
Kirchhoff,  296. 

Koenig,  192, 196,  199,  212,  217,  218. 
Kundt,  212. 
Kundt's  experiment,  210. 

Langley,  303. 

Laplace,  150. 

Law,  Hooke's,  154;  of  Boyle,  128; 
of  reflection,  241,  242;  of  re- 
fraction from  imdulatory  theory, 
258;  of  sines,  257. 

Laws  of  motion,  45. 

Length,  of  seconds  pendulum,  97, 
98;  unit  of,  4., 

Lens,  273 ;  achromatic,  289 ;  con- 
verging, 275  ;  diverging,  274  ; 
image  in,  275  ;  optical  centre  of, 
278. 

Lever,  71 ;  arm,  66 ;  by  theory  of 
work,  73, 

Light,  226;  complexity  of,  285; 
extinction  of,  327 ;  interference 
of,  297;  nature  of,  220;  polar- 
ized, 317;  propagation  of,  228; 
speed  of,  232,  239;  theory  of, 
227;  recomposition  of,  286. 

Limits  of  pitch,  225 ;  of  hearing, 
225. 

Limma,  179. 

Liquids,  density  of ,  111;  in  com- 
municating tubes,  104 ;  free  sur- 
face of,  103. 

Lissajous'  optical  method,  169. 

Longitudinal  motion,  317;  vibra- 
tions, 140. 


INDEX. 


341 


Major  chord,  177. 

Marietta's  tlask,  136 ;  law,  129. 

Mass,  42;  displacement,  43;  unit 
of,  5. 

Material  particle,  8. 

Matter,  physics  of,  6;  quantity  of, 
42. 

Maxwell,  8,  226. 

Measurement,  fundamental  units 
of,  4. 

Mechanics,  6. 

Medium  of  propagation,  139. 

Melde's  experiment,  188. 

Meniscus,  121. 

Mercury,  column  of,  127,  128 ; 
density  of,  106. 

Metre  des  archives,  4. 

Mica,  328. 

Michelson's  method,  238. 

Minimum  deviation,  268,  271. 

Minor  chords,  177,  179 ;  and  tran- 
sition, 179. 

Mixing  colored  lights,  312. 

Modulus  of  elasticity,  153 ; 
Young's,  153. 

Moment,  of  a  couple,  71;  of  a 
force,  66 ;  of  inertia,  85,  87,  88, 
89,  90;  of  the  resultant,  67; 
statical,  95. 

Momentum,  43 ;  change  of,  46. 

Monochord,  186. 

Motion,  8 ;  laws  of,  45 ;  on  inclined 
plane,  27;  rectilinear,  11 ;  simple 
harmonic,  31 ;  uniformly  accele- 
rated, 13;  uniform  circular,  29. 

Mouthpiece,  organ  pipe,  206,  212, 
213. 

Music,  182. 

Musical,  importance  of  resultant 
tones,  222;  pitch,  223;  scale, 
220;  sounds,  221. 


Narrow  apertures,  301. 

Nature  of  light,  226;  of  sound, 
137 ;  order  of  nature,  3. 

Negative  uniaxial  crystals,  321, 
322. 

Newton,  46,  285,  286,  288. 

Newtonian  explanation  of  refrac- 
tion, 259. 

Newton's  corpuscular  theory,  227 ; 
experiments,  288 ;  equation  for 
velocity,  149 ;  laws  of  motion, 
45;  rings,  333. 

Nicol's  prism,  325;  extinction  of 
light  by,  327. 

Nodes,  185,  206,  211. 

Normal,  pressure  on  curved  film, 
122;  spectrum,  304. 

Octave,  176. 

Optic  axis,  320. 

Optical  centre  of  a  lens,  278. 

Optics,  geometrical,  228  ;  physical, 

228 ;  ordinary  ray,  320,  326. 
Organ  of  hearing,  137. 
Organ  pipes,  length  of,  201. 
Oscillation,  centre  of,  96 ;  period 

of,  93. 
Overtones,  190;  beats  due  to,  213; 

relation  of  to  fundamental,  204, 

206. 

Parabola,  24,  165. 

Parallel  fprces,  68;  axis,  90. 

Parallelogram  of  velocities,  23. 

Partial  tones,  191. 

Particle,  material,  8. 

Pascal,  127. 

Pascal's  law,  100. 

Path,   9;    from  an   image  to  the 

eye,  248;  of  a  projectile,  24;  of 

rays,  245. 


342 


INDEX. 


Pendulum,  equivalent  simple,  95 ; 
ideal  simple,  92;  Rater's,  97; 
oscillation  of,  98 ;  period  of, 
93 ;  physical,  94 ;  reversibility 
of,  9G;  seconds,  length  of,  97, 
98. 

Penumbra,  232. 

Percussion,  centre  of,  96. 

Period  of  motion,  33;  of  pendu- 
lum, 93;  of  oscillation,  98. 

Periodic  function,  155. 

Perturbations  at  extremities,  212. 

Perverted  image,  230. 

Phase,  33. 

Physics,  defined,  1 ;  methods  of, 
3  ;  of  the  ether,  G  ;  of  matter,  6. 

Pitch,  176;  limits  of,  225;  musi- 
cal, 224 ;  normal,  224. 

Plane,  of  incidence,  257;  of  re- 
fraction, 257;  of  symmetry,  39, 
335;  principal,  321. 

Plane  mirror,  images  in,  243 ;  ro- 
tation of,  249. 

Plates,  transverse  vibration  of, 
197. 

Polarization,  317,  318;  angle  of, 
:'.24 ;  by  double  refraction,  323 ; 
by  reflection,  324. 

Polarized  light,  317;  circularly, 
319,  334;  colors  produced  by, 
330;  complementary  colors  by, 
330 ;  elliptically,  319  ;  plane,  318, 
335. 
Polarizer,  318,  J527. 

Positive  uniaxial  crystal,  321. 

Power,  52;  dispersive,  287,  289, 
291. 

Press,  hydraulic,  101. 

Pressure,  centre  of,  106,  108;  di- 
rectly as  depth,  103;  on  any 
immersed  surface,  105;  on  a 


horizontal  plane,  102;  on  dis- 
charging vessel,  104. 

Primary  color  sensations,  313. 

Principal  axis,  252 ;  focus,  251. 
273;  plane,  321. 

Principle  of  Archimedes,  109 ;  of 
Huyghens,  169. 

Prism,  Nicol's,  325;  refraction 
through,  267. 

Projection  upward,  17. 

Propagation,  medium  of,  139. 

Pump,  air,  130. 

Quality  of  sound,  214. 

Quartz,  double  refraction  in,  333 : 

left-handed,  334;  right-handed, 

334. 
Quincke,  116. 

Radiation,  process  of,  226. 

Radius  of  gyration,  86. 

Range  of  jets,  134;  of  projectiles, 
25. 

Rarefaction,  142. 

Ray,  extraordinary,  320;  ordinary, 
320. 

Recomposition  of  white  light,  286. 

Rectilinear  motion,  11 ;  propaga- 
tion, 228. 

Reflection  of  a  plane  wave,  171 ; 
law  of,  241 ;  law  of  from  undu- 
latory  theory,  242;  polarization 
by,  324 ;  total  internal,  263,  264, 
325. 

Refracted  ray,  257,  262. 

Refraction,  257;  at  a  plane  sur- 
face, 260 ;'  at  a  spherical  surface, 
271;  double,  319,  321;  index  of, 
258 ;  Newtonian  explanation  of, 
259;  of  a  spherical  wave,  266; 
through  a  prism,  267. 


INDEX. 


343 


Refractive  index,  325,  326. 

Refrangibility,  285;  order  of,  286. 

Regnault,  145. 

Relative  index  of  refraction,  258, 
326. 

Representation,  graphical  of 
work,  52. 

Resolution  in  rectangular  direc- 
tions, 22. 

Resonance,  200. 

Resonators,  201. 

Restitution,  forces  of,  154. 

Resultant,  18,  72;  of  two  simple 
harmonic  motions,  36;  to  find, 
21. 

Resultant  tones,  218;  musical  im- 
portance of,  222. 

Reversal  of  lines,  296. 

Rigidity  of  a  cord,  184. 

Rings,  colored,  332;  iridescent, 
333 ;  Newton's,  333. 

Rods,  transverse  vibration  of,  193. 

Roeraer,  233. 

Rotation  of  analyzer,  329 ;  .  of 
plane  of  polarization,  334. 

Rowland,  305,  307. 

Saccharimeter,  335. 

Scales,   musical,    220;    tempered, 

180. 

Second,  definition  of,  6. 
Secondary  axes,  280,  281. 
Seconds  pendulum,  93,  97. 
Segmental  vibrations,  186. 
Selenite,  328. 
Sensation,  of  light,  226 ;  of  sound, 

137. 

Sensibility  of  the  balance,  75. 
Shadows,  theory  of,  231. 
Shoaling,  261. 
Simple  harmonic  motion,  31 ;  ap- 


plied to  sound,    153;   equations 

of,  47. 

Sines,  curve  of,  154,  156,  157. 
Sinusoid,  156. 
Siphon,  135. 
Soap  bubble,  123. 
Sound,   and  hearing,  137;    defini- 
tion  of,    140;    source   of,    138; 

transmission  of,   140.- 
Specific  gravity,   110;    heat,   151; 

of  solids,  111. 
Spectra,  absorption,    295 ;   bright 

line,  294;  continuous,  295. 
Spectrometer,  293. 
Spectroscope,  293. 
Spectrum,    286;    diffraction,    303; 

normal,  304;  solar,  293. 
Speed  and  velocity,  10 ;   of  light, 

169,  232,  239. 
Spherical,    aberration,    254,    282; 

mirrors,  285. 

Statical  friction  in  fluids,  99. 
Statics,  7. 

Stationary  waves,  185. 
Strings,  transverse   vibration  of, 

182. 

Subjective  colors,  315. 
Successive  reflection,  245. 
Summational  tones,  218. 
Superficial  viscosity,  125. 
Surface,   aplanatic,    283 ;    caustic, 

253,  261;  tension,  116. 
Surface  tension,  116;  energy  and, 

120;  illustrations  of ,  119. 
Suspension,  axis  of,  94. 
Symmetry,  plane  of,  39,  335. 

Tait,  64,  226. 

Temperament,  equal,  181. 
Tempered  scales,  180. 
Tension,  surface,  116. 


344 


INDEX. 


Theory,  of  double  refraction,  321 ; 

physical  theory  of  music,  176. 
Thin  films,  299. 
Third,  interval  of,  176. 
Time,  interval,  33;   unit  of,  6. 
Tones,  partial,  191 ;  resultant,  218. 
Torricelli,  126;  theorem,  133. 
Total  reflection,  263,  264,  325. 
Tourmaline,  317,  325. 
Transformations  of  energy,  61. 
Transverse    vibration,     140;      of 

plates,    197;  of    rods,    193;    of 

strings,  182. 

Triangle  of  motions,  20. 
Tuning-fork,  195. 
Tyndall,  174. 

Umbra,  231. 

Undulatory  theory,    228;    law   of 

refraction  from,  258. 
Uniaxial   crystals,  negative,  321 : 

positive,  321. 
Unit,  of  length,  4;   of  mass,  5;  of 

power,  52 ;  of  time,  6 ;  of  work, 

52. 
Upper  partials,  191. 


Velocity,  angular,  84 ;  at  any  point, 

34 ;  linear,  10,  85. 
Velocity    of    sound,  experimental 

determination  of,  144 ;  in  solids, 

152;    theoretical   determination 

of,  146. 
Vibration,  frequency,  153,  306 ;  of 

strings  in  segments,  184. 
Virtual  image,  244,  275,  276,  281, 

282 ;  point-source,  272. 
Viscosity,  99 ;  superficial,  125. 
Volume,  42. 

Watt,  52. 

Wave,  142. 

Wave-front,  143. 

Wave-lengths  of  light,  306. 

Wertheim,  153,  213. 

White  light,  complexity  of,  285 ; 
recomposition  of,  286. 

Wollaston,  293. 

Work,  defined,  51;  graphical  rep- 
resentation of ,  52;  unit  of,  52. 

Young,  313. 

Young's  modulus,  153. 


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